Sunday, 9 January 2022

EQUATION OF STRAIGHT LINE(X)

            
EXERCISE--1
                  -------------------..


1) Find the slope of a line whose inclination is:
a) 30°. 1/√3
b) 90°. not defined

2) Find the inclination of a line whose slope is
a) √3.           60°
b) 1/√3.        30°
c) 1.             45°  


3) Find the slope of a line which passes through the points:
a) (0,0) and (4, -2). -1/2
b) (0,-3) and (2, 1). 2
c) (2,5) and (-4, -4). 3/2
d) (-2,3) and (4, -6). -3/2

4) Show that the following points are collinear:
a) (1,5),(3,14),(-1,-4).
b) (3,-4),(1,2),(2,1)
c) (3,-2),(-1,1),(-5,4)
d) (4,7),(-2,-5),(2,3)


5) If the slope of the line joining the points A(x,2) and B (6, -8) is -5/4, find the value of x. -2

6) Find y if the slope of the line joining (-8,11),(2,y) is -4/3. -7/3

7) Show that the line through the points (5,6) and (2,3) is parallel to the line through the points (9,-2) and (6,-5).     

8) Find the value of x so that the line through (3,x) and (2,7) is parallel to the line through (-1,4) and (0,6). 9 

9) Show that the line through the point (-2,6) and (4,8) is perpendicular to the line through the points (3,-3) and (5,-9).   

10) If A(2,-5), B(-2,5), C(x,3) and D(1,1) be four points such that a AB and CD are perpendicular to each other, find the value of x. 6

11) Without using Pythagora's theorem, show that the points
a) A(1,2), B(4,5) and C(6,3) 
b) A(0,4), B(1,2) and C(3,3) are the vertices of a right-angled triangle.

12) Using slopes, find the value of x for which the points A(5,1), B(1,-1) and C(x,4) are collinear. 11

13) Find the value of x for which the points (x, -1),(2,1) and (4,5) are collinear. 1

14) Using slopes, show that point
a) A(-4,-1), B(-2,-4), C(4,0) and D(2,3)
b) (-4,-1),(-2,-4),(4,0),(2,3) 
 are taken in order, are the vertices of a rectangle.  

15) Using slopes prove that the points A(-2,-1), B(1,0), C(4,3) and D(1,2) are the vertices of a parallelogram.

16) If the points A(a,0), B(0,b) and P(x,y) are collinear, using slopes, prove that x/a + y/b = 1.

17) If three A(h,0), P(a,b) and B(0,k) lie on a line, show that: a/h + b/k = 1.

18) The vertices of a quadrilateral are A(-4,2), B(2,6), C(8,5) and D(9,-7). Using slopes, show that the midpoints of the sides of the quadrilateral ABCD form a parallelogram.

19) Show that the points A(0,6), B(2,1) and C(7,3) are three corners of a square ABCD. find
a) the co-ordinates of the fourth vertex D. (5,8) 

22) A(1,1), B(7,3) and C(3,6) are the vertices of a ∆ABC. If D is the midpoint of the BC and AL perpendicular to BC. find the Slope of 
A) AD. 7/8
B) AL. 4/3

23) The slope of a line is double of the slope of another line. If tangents of the angle between them is 1/3, find the slope of the other line. 1, 1/2

24) A quadrilateral has vertices (4,1),(1,7),(-6,0) and (-1,-9). Show that the midpoints of the sides of this quadrilateral form a parallelogram.

               EXERCISE -2
                 ----------------

1) Find the equation of a line parallel to the x-axis at a distance of 
a) 4 units above it. y-4= 0
b) 5 units below it. y+5= 0

2) Find the equation of a line parallel to the y-axis at a distance of 
a) 6 units to its right. x- 6= 0
b) 3 units to the left. x+ 3= 0

3)a) Find the equation of a line parallel to the x-axis and having intercept -3 on y-axis. y+3= 0

b) Find the equation of the line parallel to x-axis and having intercept -2 on y-axis. y= -2

c) Find the equation of the line parallel to x-axis and passing through (3,-5). y=-5

d) Find the equation of the line perpendicular to x-axis and having intercept -2 on x-axis. x= -2


4) Find the equation of a horizontal line passing through the point (4,-2). y+2 = 0

5)a) Find the equation of a vertical line passing through the point (-5, 6). x+5= 0

b) Find the equation of the straight lines which pass through (4,3) and are respectively parallel and perpendicular to the x-axis. y=3, x=4

6)a) Find the equation of a line which is equidistant from the lines x= -2 and x= 6. x= 2

b) Find the equation of a line equidistant from the lines y= 10 and y= -2. y= 4

7) Find the equation of a line which is equidistant from the line y= 8 and y= -2. y= 3

8) Find equation of a line
a) whose slope is 4 and which passes through the point (5,-7). 4x - y- 27= 0 
b) Whose slope is -3 and which passes through the point (-2,3). 3x+ y+3= 0


9) Find the equation of a line whose inclination with the x-axis is 30° and which passes through the point (0,5). x - √3 y + 5√3 = 0

10) find the equation of a line which cuts of intercept 5 on the x-axis and makes an angle 60° with the positive direction of x-axis. √3 x - y -5√3= 0

13) Find the equation of the line passing through the point P(4,-5) and parallel to the line joining the points A(3,7) and B (-2,4).

14) find the equation of the line passing through the point P(-3,5) and perpendicular to the line passing through the points A(2,5) and B (-3,6). 5x - y+20= 0 

15) find the slope and the equation of the line passing through the points
A) (3,-2) and (-5,-7). 5/8, 5x- 8y-31= 0
B) (-1,1) and (2,-4). -5/3, 5x+ 3y+2= 0
C) (5,3) and (-5,-3). 3/5, 3x- 5y= 0
D) (a,b) and (-a,b). 0, y= b

16) Find the angle which the line joining the points (1,√3) and (√2, √6) makes with the x-axis. 60°

17) Prove that the points A(1,4),.B(3,-2) and C (4,5) are collinear. also find the equation of the line on which these points lie. 3x+y= 7

18) If A(0, 0) B(2,4) and C(6,4) are the vertices of a ∆ABC, find the equations of its sides. y= 4, 2x-3y= 0, 2x -y= 0.

19) If A(-1,6), B(-3,-9) and C(5,-8) are the vertices of a ∆ ABC, find the equation of its medians. 29x+4y+5= 0, 8x- 5y-21, 13x+ 14y+ 47= 0.

20) Find the equation of the perpendicular bisector of the line segment whose end points are A(10,4) and B(-4,9). 28x - 10y -19= 0
 
21) Find the equations of the altitudes of a ∆ABC, Whose vertices are A(2,-2), B(1,1) and C(-1,0). 2x+y -2= 0, 3x- 2y- 1= 0, x- 3y -1= 0

22) If A(4,3), B(0,0) and C (2,3) are the vertices of a ∆ABC, find the equation of the bisector of angle A. x- 3y+5= 0

23) The midpoints of the sides BC, CA and AB of a ∆ ABC are D(2,1), E(-5,7) and F(-5,-5) respectively. Find the equations of the sides of if 1423 are the third season of ∆ABC. x- 2= 0, 6x - 7y +79= 0, 6x+ 7y +65= 0

24) If A(1,4), B(2,-3) and C(-1,-2) are the vertices of a∆ ABC, find the equation of
a) the median through A.. 13x- y -9= 0
b) the altitude through A. 3x-y+1= 0
c) the perpendicular bisector of BC. 3x- y - 4= 0

             
                  EXERCISE -3
                    ---------------


1) Find the equation of the line which makes an angle of 30° with the positive direction of the x-axis and cuts off an intercept of 4 units with the negative direction of the y-axis. x - √3y - 4√3= 0

2) Find the equation of a straight line :
a) with slope 2 and y-intercept 3. y= 2x+3
b) with slope -1/3 and y-intercept - 4. x+ 3y+12= 0
c) with slop -2 and intersecting x- axis at a distance of 3 units to the left of the origin. 2x+ y+6= 0
d) Slope= 3, y-intercept=5. 3x-y+5= 0
e) Slope=-1, y-intercept= 4. x+y-4= 0
f) Slope= -2/5, y-intercept= -3. 2x+ 5y+15= 0


3) Find the equation of the line cutting off an intercept -2 from the y-axis and inclined to the axes. x- y - 2= 0, or x+ y +2= 0

4) Find the equation of a line that has y-intercept -4 and is parallel to the line joining (2,-5) and (1,2). 7x + y+4= 0

5) Find the equation of the line through the point (-1,5) and making an intercept of -2 on the y-axis. 7x+ y + 2= 0

6) find the equation of a line which is perpendicular to the line joining (4,2) and (3,5) and cuts off an intercept of length 3 on y- axis. x - 3y+9= 0

7) find the equation of the perpendicular to the line segment joining (4,3) and (-1,1) if cuts off an intercept -3 from y-axis. 5x+ 2y+ 6= 0

8) find the equation of the straight line intersecting y-axis at a distance of 2 units above the origin and making an angle 30° with the positive direction of the x-axis. x - √3 y+ 2√3= 0

9) Find the equation of the line which is parallel to the line 2x - 3y = 8 and whose y-intercept is 5 units. 2x- 3y +15= 0

10) Find the equation of the line which is perpendicular to the line x - 2y = -5 and passing through (0,3). 2x + y -3= 0     

11) Find the equation of the line passing through the point (2,3) and perpendicular to the line 4x +3y= 10. 3x- 4y +6= 0

12) Find the equation of the line passing through the point (2,4) and perpendicular to the x-axis. x = 2

13) Find the equation of the line that has x-intercept -3 and which is perpendicular to the line 3x +5y= 4. 5x- 3y + 15= 0

14) Find the equation of the line passing through the midpoint point of the line joining the point (6,4) and (4, -2) and perpendicular to the line 3x +2y= 8. 2x- 3y -7= 0

15) Find the equation of the line whose y-intercept is -3 and which is perpendicular to the line joining the points (-2,3) and (4,-5). 3x- 4y -12= 0

16) Find the equation of the line passing through (-3,5) and perpendicular to the line through the points (2,5) and (-3,6). 5x - y+20= 0

17) A line perpendicular to the line segment joining the points (1,0) and (2,3) divides it in the ratio 1:2. Find the equation of the line. 3x+ 9y-13= 0


MISCELLANEOUS-1 (A)
-------------------------------

1) Find the gradient of the line joining the pair of points:
a) (0,3),(4,5). 1/2
b) (0,-2),(-2,4). -3
c) (1/2,3/2),(5/2,7/2). 1
d) (√3+1,2),(√3+3,4). 1

2) Find the Slope of a line through each of the pair of the following points:
a) (2,3) and (3,4). 1
b) (2,-1) and (4,1). 1
c) (-3,-2) and (-2,-1). 1
d) (-5,-3) and (4,3). 2/3

3) Find the inclination of the line joining the pair of points:
a) (1,2),(2,3). 45°
b) (-1,-3),(3,1). 45°


4) If A(4,-3), B(6,5) and C(5,1) are three points, find the slope of AB and BC. Hence show that the points are collinear. 4, 4

5) Show that the following points are collinear:
a) A(5,-2), B(4,-1) and C(1,2). 

6)a) If (-5,a),(3,6) and (7,8) are collinear, find a. 2

b) If A(2,a), B(3,-1) and C(4, -5) are collinear, find a. 3

c) Find a, if the points (-2,3),(3,4),(a,5) are collinear. 8

d) If the points (a,1),(1,2) and (0, b+1) are collinear, Show that 1/a + 1/b = 1.

7)a) If 2y - p²x= 3 and 2y - 4px +1= 0 are parallel, find the value of p. 4

b) If y - 2x = 3 and 2y = px +8 are parallel, find the value of p. 4

c) If 3(k -1)y - 6x = 2 and 4y - 8x +10= 0 are parallel, find the value of k. 2

8)a) If (p+1)x + y= 3, and 3y - (p- 1)x= 4 are perpendicular, Find the value of p. ±2

b) If 2my - 3x= 4 and 3my + 8x =10 are perpendicular to each other, find m. ±2

c) If y+ (2p+1)x +3= 0 and 8y- (2p -1)x= 5 are perpendicular, find the value of p. ±3/2

9) State the slope (m) and y-intercept (c) of the line 2y = 4x - 3. 2, -3/2

10) Find the slope of a line
a) parallel 
b) perpendicular to to 2y= 3x+1.
     3/2, -2/3


MISCELLANEOUS-- 1(B)
-----------------------------

1) State the slope(m) and y-intercept(c) of the line:
a) y= x+1. 1,1
b) 2y= 4x+1. 2,1/2
c) 3y= 6x -2. 2, -2/3
d) 2y= - 3x -4. -3/1, 2

2) Find the Slope of a line which is parallel to:
a) 2y= x+2. 1/2,
b) 3y/4= 2x -2. 8/3


3) Find the Slope of a line which is perpendicular to:
a) 3y= x+5. -3
b) 2y/3= x -3. -2/3

4)a) Find the equation of the line y-intercept of 3, and a slope of 2. y= 2x+ 3.      

b) Find the equation of the line y-intercept of 7, and a slope of 2. y= 2x+ 7.      

c) Find the equation of the line with y-intercept of 4 and a slope of -3. y= - 3x +4.      

d) Find the equation of the line with y-intercept of 1/2 and a slope of 2. 2y= 4x+1.     

e) Find the equation of the line with y-intercept of 5, and a slope of -2/3. 3 y= - 2x+15.      

f) Given, y-intercept of 2, and an inclination of 45° with the positive direction of the x-axis. y= x+ 2.      

g) Find the equation of the line having y-intercept of 4 and which is equally inclined to the axes, in the second quadrant. y= x+ 4

5)a) Find the equation of a line through (0,3) and having slope = 4. y= 4x +3

b) Find the equation of a line through (1,2) and having slope = 3. y= 3x - 1


6)a) Find the equation of the line through (0,2) and parallel to y= 3x +2. y= - x+ 3.      

b) Find the equation of the line through (4,0) and parallel to 3y= 6x +2. y= 2x - 8     
c) Find the equation of the line through (0,3) and parallel to 2y= x - 2. 2y= x -6

7) Find the equation of the line through (0,2) and perpendicular to
a) y= 2x+3. 2y= - x+4.     
 
b) y= x/3 +2. y= - 3x+2

c) Find the equation of the line through (0,3) and perpendicular to 2y= x + 1. y= - 2x+ 3.      


8)a) Find the equation of the line through (0,2) and (-2,0). y= x+2

b) Find the equation of the line through (3,0) and (0,3). y= -x+3

c) Find the equation of the line through (2,1) and (4,3). y= x -1

9)a) Find the equation of the line through (2,3) and (3,4) and y-intercept of 5 units. y= x+ 5

b) Find the slope of the line joining the points (3,4) and (0,16). Hence or otherwise, write down the equation of this line. -4, y= - 4x+16

10) The equation of a line is y= 3x -5. Write down the slope of this line and intercept made by it on the y-axis. Hence or otherwise, write down the equation of a line which is parallel to this line and which passes through the point (0,5). 3, -5, y= 3x+5 

11) A(2,1), B(5,3), C(-1,3) are the vertices of the triangle ABC. Find
a) equation of the median AD. 2= x
b) the equation of the altitude BE. 2y= 3x -9     
c) the equation of the altitude CF. 2y= -3 x+ 3      

12) find the equation of the line through (-4,8) and parallel to x-axis. y= 8

13) find the equation of the line through (3,5) and perpendicular to the axis. y= 5

14) A(1,3) and C(6,8) are the opposite vertices of a square ABCD. Find the equation of the diagonal BD. x +y = 9

15) Find the equation of the perpendicular bisector of the line segment joining (3,2) and (7,6). x+ y = 9

16) A(-1,4) and B(5, -2) are two points. Find the equation of the perpendicular bisector of AB. y + x -3= 0 

17) A straight line cuts off, on the axes of co-ordinates positive intercepts whose sum is 7. If the line passes through the point (-3,8), find its equation. 4x +3y= 12

18) A straight line cuts off, on the axes, positive intercepts whose sum is 7. If the line passes through the point (-8, 9), find its equation. 3x +4y= 12

19) The coordinates of the vertex A of a square ABCD are (1,2) and the equation of the diagonal BD is x + 2y= 10. Find the equation of the diagonal and the coordinates of the centre of the square. y- 2x= 0; (2,4)

20) A, B are the points (0,6) and (10,0). O is origin, OM is a median and OP an altitude of triangle AOB. Find the equation of OM and OP. 5y- 3x= 0; 3y- 5x= 0

21) Find the equation of the line passing through the point (3,4) such that the portion between the axes is divided by P in the ratio 2:3. y + 2x= 10

22) Find the equation of a line, which has the y-intercept of 5 and is parallel to the line 4x - 6y= 9. Find the coordinates of the point, where it cuts the x axis. 3y- 2x= 15, (-7.5,0)



             
         













   EXERCISE--1
                ----------------


1) Find the gradient of the line joining the pair of points:
a) (0,3),(4,5).                                 1/2
b) (0,-2),(-2,4).                                 -3
c) (1/2,3/2),(5/2,7/2).                     1
d) (√3+1,2),(√3+3,4).                       1

2) Find the Slope of a line through each of the pair of the following points:
a) (2,3) and (3,4).                              1
b) (2,-1) and (4,1).                             1
c) (-3,-2) and (-2,-1).                          1
d) (-5,-3) and (4,3).                        2/3

3) Find the inclination of the line joining the pair of points:
a) (1,2),(2,3).                                  45°
b) (-1,-3),(3,1).                                45°


4) If A(4,-3), B(6,5) and C(5,1) are three points, find the slope of AB and BC. Hence show that the points are collinear.                                  4, 4

5) Show that the following points are collinear:
a) A(5,-2), B(4,-1) and C(1,2). Hence find the inclination of the line AC.              135°

6)a) If (-5,a),(3,6) and (7,8) are collinear, find a.                                 2

b) If A(2,a), B(3,-1) and C(4, -5) are collinear, find a.                                3

c) Find a, if the points (-2,3),(3,4),(a,5) are collinear.                             8

d) If the points (a,1),(1,2) and (0, b+1) are collinear, Show that 1/a + 1/b = 1.

7)a) If 2y - p²x= 3 and 2y - 4px +1= 0 are parallel, find the value of p.      4

b) If y - 2x = 3 and 2y = px +8 are parallel, find the value of p.             4

c) If 3(k -1)y - 6x = 2 and 4y - 8x +10= 0 are parallel, find the value of k.                                                        2

8)a) If (p+1)x + y= 3, and 3y - (p- 1)x= 4 are perpendicular, Find the value of p.                                        ±2

b) If 2my - 3x= 4 and 3my + 8x =10 are perpendicular to each other, find m.                                                     ±2

c) If y+ (2p+1)x +3= 0 and 8y- (2p -1)x= 5 are perpendicular, find the value of p.                                    ±3/2

9) State the slope (m) and y-intercept (c) of the line 2y = 4x - 3.              2, -3/3

10) Find the slope of a line
a) parallel 
b) perpendicular to to 2y= 3x+1. 
     3/2, -2/3


               EXERCISE --2
                 -----------------
1) State the slope(m) and y-intercept(c) of the line:
a) y= x+1.                                      1,1
b) 2y= 4x+1.                               2,1/2
c) 3y= 6x -2.                             2, -2/3
d) 2y= - 3x -4.                           -3/1, 2

2) Find the Slope of a line which is parallel to:
a) 2y= x+2.                                    1/2,
b) 3y/4= 2x -2.                              8/3


3) Find the Slope of a line which is perpendicular to:
a) 3y= x+5.                                      -3
b) 2y/3= x -3.                                -2/3

4)a) Find the equation of the line y-intercept of 3, and a slope of 2.   y= 2x+ 3.      

b) Find the equation of the line y-intercept of 7, and a slope of 2.   y= 2x+ 7.      

c) Find the equation of the line with y-intercept of 4 and a slope of -3.       y= - 3x +4.      

d) Find the equation of the line with y-intercept of 1/2 and a slope of 2.      2y= 4x+1.     

e) Find the equation of the line with y-intercept of 5, and a slope of -2/3.    3 y= - 2x+15.      

f) Given, y-intercept of 2, and an inclination of 45° with the positive direction of the x-axis.       y= x+ 2.      

g) Find the equation of the line having y-intercept of 4 and which is equally inclined to the axes, in the second quadrant.                 y= x+ 4

5)a) Find the equation of a line through (0,3) and having slope = 4.    y= 4x +3

b) Find the equation of a line through (1,2) and having slope = 3.    y= 3x - 1


6)a) Find the equation of the line through (0,2) and parallel to y= 3x +2.                   y= - x+ 3.      

b) Find the equation of the line through (4,0) and parallel to 3y= 6x +2.                                        y= 2x - 8     
c) Find the equation of the line through (0,3) and parallel to 2y= x - 2.                                             2y= x -6

7) Find the equation of the line through (0,2) and perpendicular to
a) y= 2x+3.                   2y= - x+4.     
 
b) y= x/3 +2.                       y= - 3x+2

c) Find the equation of the line through (0,3) and perpendicular to 2y= x + 1.                     y= - 2x+ 3.      


8)a) Find the equation of the line through (0,2) and (-2,0).       y= x+2

b) Find the equation of the line through (3,0) and (0,3).        y= -x+3

c) Find the equation of the line through (2,1) and (4,3).        y= x -1

9)a) Find the equation of the line through (2,3) and (3,4) and y-intercept of 5 units.               y= x+ 5

b) Find the slope of the line joining the points (3,4) and (0,16). Hence or otherwise, write down the equation of this line.                 -4, y= - 4x+16

10) The equation of a line is y= 3x -5. Write down the slope of this line and intercept made by it on the y-axis. Hence or otherwise, write down the equation of a line which is parallel to this line and which passes through the point (0,5).     3, -5, y= 3x+5 

11) A(2,1), B(5,3), C(-1,3) are the vertices of the triangle ABC. Find
a) equation of the median AD.   2= x
b) the equation of the altitude BE.  2y= 3x -9     
c) the equation of the altitude CF.  2y= -3 x+ 3      

12) find the equation of the line through (-4,8) and parallel to x-axis.       y= 8

13) find the equation of the line through (3,5) and perpendicular to the axis.                                        y= 5

14) A(1,3) and C(6,8) are the opposite vertices of a square ABCD. Find the equation of the diagonal BD.                                    x +y = 9

15) Find the equation of the perpendicular bisector of the line segment joining (3,2) and (7,6).    x+ y = 9

16) A(-1,4) and B(5, -2) are two points. Find the equation of the perpendicular bisector of AB.    y= x - 1.   

17) A straight line cuts off, on the axes of co-ordinates positive intercepts whose sum is 7. If the line passes through the point (-3,8), find its equation.         4x +3y= 12

18) A straight line cuts off, on the axes, positive intercepts whose sum is 7. If the line passes through the point (-8, 9), find its equation.   3x +4y= 12

19) The coordinates of the vertex A of a square ABCD are (1,2) and the equation of the diagonal BD is x + 2y= 10. Find the equation of the diagonal and the coordinates of the centre of the square.     y- 2x= 0; (2,4)

20) A, B are the points (0,6) and (10,0). O is origin, OM is a median and OP an altitude of triangle AOB. Find the equation of OM and OP.         5y- 3x= 0; 3y- 5x= 0

21) Find the equation of the line passing through the point (3,4) such that the portion between the axes is divided by P in the ratio 2:3.      y + 2x= 10

22) Find the equation of a line, which has the y-intercept of 5 and is parallel to the line 4x - 6y= 9. Find the coordinates of the point, where it cuts the x axis.              3y- 2x= 15, (-7.5,0)
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Friday, 7 January 2022

INTEGRATION BY PARTS

            EXERCISE-1
             -----------------

1) ∫ x eˣ.                                 (x-1)eˣ 

2) ∫ x e²ˣ.                     (x/2 - 1/4) e²ˣ 

3) ∫ x² eˣ.                       x² eˣ - 2[x eˣ]

4)  ∫ x² e⁻ˣ.                  -e-ˣ(x² + 2x+2)

5) ∫ x³ eˣ.               (x³- 3x²+ 6x -6) eˣ

6) ∫ 2x³ ₑx².                         ₑx². (x²-1)

7) ∫ log x.                            x log x - x

8) ∫ (log x)².                        x(log x)² - 2(x log x - x)

9) ∫ log (x+1).            x log(x+1) - x + log(x+1)

10) ∫ x log x.              x²/2 logx - x²/4

11) ∫ x³ log 2x.              x⁴/4 log 2x - x⁴/16

12) ∫ x³ log x.                 x⁴/4  log x - x⁴/16

13) ∫ (log(log x))/x.        log x{log(log x) -1}

14) ∫ (logx)/x².            -1/x (1+ logx)

15) ∫ log(1+ x²).         x log(x²+1) - 2x + 2 tan⁻¹x

16) ∫ x log(1+ x).         x²/2 log(x+1) - 1/2[x²/2 - x + log|x+1|]

17) ∫ (log x)² x.           x²/2 ((logx)² - log x + 1/2)

18) ∫ xⁿ  log x.                  xⁿ⁺¹/(n+1) . Log x - xⁿ⁺¹/(n+1)²

19) ∫ (log x/xⁿ          x¹⁻ⁿ/(1 - n) . Log x - x¹⁻ⁿ/(1- n)²

20) ∫ log₁₀x.             1/log 10  [x(log x - 1)]

21) ∫ x sin 2x.                -x/2 cos 2x + 1/4 sin 2x

22) ∫ x sin 3x dx.             -x/3 cos 3x + 1/9 sin 3x

23) ∫ x sin²x.            x²/4 - x/4 sin 2x - 1/8 cos 2x


24) ∫ x² sinx.                -x² cos x +2|x sin x + cos x|

25) ∫ sin√x.                 2(- √x cos√x + sin√x)

26) ∫ x² sin²x.                x³/6 - x²/4 sin 2x  -  x/4 cos 2x + 1/8 sin 2x

27) ∫ x sin³x.                  -3x/4 cosx + 3/4 sinx + x/12 cos 3x - 1/36 sin 3x 

28) ∫ sin³√x.              3√x³ sin³√x + 6 ³√x cos³√x + 6 cos³ √x

29) ∫ x cos x.                 x sinx + cosx

30) ∫ x² cos x.          x² sinx + 2x cosx - 2 sinx

31) ∫ x² cos 2x.              x²/2 sin2x + x/2 cos 2x -1/4 sin 2x

32) ∫ x cos²x.                x²/4 + (x sin 2x)/4 + (cos 2x)/8

33) ∫ x³ cos x².               x²/2 sin x² + 1/2 cos x²

34) ∫ cos√x.                     2(√x sin√x + cos√x)

35) ∫ x cos³x.               x/12 sin3x + 1/36 cos 3x + 3x/4 sinx + 3/4 cos x

36) ∫ x sec²x.           x tanx + log|Cos x|

37) ∫ sec³x.           1/2 sec tanx + 1/2 log|secx + tanx|

38) ∫ x cosec²x                  - x cot x + log|sinx|

39) ∫ cosec³x.         -1/2 cosecx cot x + 1/2 log|tan(x/2)|


40) ∫ x tan²x            x tanx - log secx - x²/2

41) ∫ sin⁻¹x.          x sin⁻¹x + √(1- x²)

42)  ∫ sin⁻¹(3x - 4x³).        3x sin⁻¹x + 3√(1- x²)

43)  ∫ sin⁻¹{2x/(1+ x²)}.       2x tan⁻¹x - log|1+ x²|

44) ∫ x sin⁻¹x.          x²/2 sin⁻¹x + x/4 √(1- x²) - 1/4 sin⁻¹x

45) ∫ x² sin⁻¹x.             x³/3  sin⁻¹x + 1/3 √(1- x²) - 1/9 √(1-x²)³

46) ∫ sin⁻¹√{x/(a+x)}.                       x tan⁻¹√(x/a) - √(ax) + a tan⁻¹√(x/a)

47) ∫ cos⁻¹(4x³ - 3x).        3x cos⁻¹x - 3 √(1- x²)

48) ∫ cos⁻¹{(1- x²)/(1+ x²)}.          2x tan⁻¹x - log|1+ x²|

49) ∫ tan⁻¹x.         x tan⁻¹x - 1/2 log|1-+x²|

50)  ∫ tan⁻¹{3x - x³)/(1- 3x²)}.       3x tan⁻¹x - 3/2 log|x²+1|

51) ∫ sec⁻¹x.          x(sec⁻¹x) - log|x + √(x²- 1)|

52) ∫ x tan⁻¹x.                  x²/2 tan⁻¹x - 1/2(x -  tan⁻¹x)   

53) ∫ x² tan⁻¹x.              x³/3 tan⁻¹x - x²/6 + 1/6 log|x²+ 1|

54) ∫ tan⁻¹{2x/(1- x²)}.        2x tan⁻¹x - log|(1+ x²)|

55) ∫ tan⁻¹ √{(1-x)/(1+x)}.          x/2 (cos⁻¹x) - 1/2 √(1- x²)

56) ∫x cot⁻¹x.        x²/2 cot⁻¹x + 1/2(x - tan⁻¹x)   


MISCELLANEOUS-1
------------------------------

1) ∫ (x - sinx)/(1- cosx).                 - x cot(x/2)

2) ∫ (x + sinx)/(1+ cosx).                 x tan(x/2)

3) ∫ (sin⁻¹x)/√(1- x²)³. x/√(1-x²)  sin⁻¹x + 1/2 log|1- x²|

4) ∫ x(sin⁻¹x)/√(1- x²).         - √(1-x²) sin⁻¹x + x

5) ∫ x²(tan⁻¹x)/(1+ x²).        x tan⁻¹x - 1/2 log|1+ x²| - 1/2 (tan⁻¹x)²

6) ** ∫ (sin⁻¹√x - cos⁻¹√x)/(sin⁻¹√x + cos⁻¹√x).         2/π [√(x - x²) - (1- 2x) sin⁻¹√x] - x

7) ∫ [√(x²+1){log(x²+1) - 2 log x}]/x⁴.    -1/3  √(1+ 1/x²)³ {log(1+ 1/x²) - 2/3}

8) ∫ x²/(x sinx + cos x).          (sinx - x cos x)/(x sinx + cosx)  

9) ∫ x sinx cosx.            -x/4 cos 2x + 1/8 sin 2x

10) ∫ sin x log(cos x).        cosx(1- log cos x)

11) ∫ ₑ√x.                      2ₑ√x . (√x -1)

12) ∫ {log(x+2)}/(x+2)².        - 1/(x+2) - {log(x+2)}/(x+2)

13) ∫ x{(sec 2x -1)/(sec 2x +1)}.    x tanx - log secx - x²/2

14) ∫ (x+1) eˣ log(xeˣ).     W    x eˣ(log (xeˣ) - 1)

15) ∫ (e ˡᵒᵍ ˣ + sinx).      x²/2 - cos x 

16) ∫ (x tan⁻¹x)/√(1+x²)³.      -Tan⁻¹x/√(1+ x²) + x/√(1+ x²)

17) ∫ (x³ sin⁻¹x²)/√(1- x⁴).     1/2 {x² - √(1- x⁴) sin⁻¹x²}

18) ∫ (x³ tan⁻¹x).       {(x⁴-1)/4}Tan⁻¹x - x³/12 + x/4.

19) ∫ tan⁻¹(√x)       (x+1)Tan⁻¹√x  - √x

20) x sinx cos 2x.         1/2[- x/3 cos 3x + 1/9 sin3x + x cosx - sinx]

21) ∫ (tan⁻¹x²).         x²/2 Tan⁻¹x² - 1/4 log|(1+ x⁴)|

22) ∫ log(1+ x²).            x log(1+ x²) - 2x + 2 tan⁻¹x

23) ∫ x² tan⁻¹x.                  x³/3 tan⁻¹x - x²/6 + 1/6 log|x²+1|

24) ∫ sec⁻¹√x.             x sec⁻¹√x - √(x -1)

25) ∫ {x² sin⁻¹x}/√(1- x²)³.      {x sin⁻¹√x}/√(1- x²) - 1/2  (sin⁻¹x)²+ 1/2 log|(1- x²)|

26) ∫ (sin⁻¹x))x².         (sin⁻¹x))x + log|{1- √(1- x²)}/x| 

27) ∫ (x+1) log x.              (x + x²/2)logx - (x + x²/4)

28) ∫ f(x) g"(x) - f"(x) g(x).            f(x) g'(x) - g(x) f'(x)+ C



                  EXERCISE- 2
                   --------------

1) ∫ eˣ(1/x - 1/x²).                        eˣ/x

2) ∫ eˣ(1/x² - 2/x³).                     eˣ/x²

3) ∫ eˣ{(x-1)/2x²}.                       eˣ/2x

4) ∫ eˣ{x/(x+1)²}.                  eˣ/(x+1)

5) ∫ eˣ {(x - 1)/(x+1)³}.        eˣ/(x+1)²

6)  ∫ eˣ {(1 - x)²/(x²+1)²}.       eˣ/(x²+1)

7) ∫ eˣ {(1+ x)/(x+2)²}.         eˣ/(x+2)

8) ∫ eˣ {(2- x)/(1 - x)²}.         eˣ/(1 -x)

9) ∫ eˣ {(x²+1)/(x+1)²}.                eˣ - eˣ/(x +1)

10) ∫ eˣ {(x -3))/(x- 1)³}.         eˣ/(x -1)²

11) ∫ eˣ(log x + 1/x).             eˣ logx

12) ∫ eˣ(log x + 1/x²).                   eˣ (logx -1/x)

13) ∫ eˣ/x {x(log x)² + 2 log x}).                    eˣ (logx)²

14) ∫  {1/(log x) - 1/(log x)².     x/logx

15) ∫ log x/(1+ log x)².            x/(logx +1)

16) ∫ log(logx) + 1/(logx)².             x log(logx) - x/(logx)


17) ∫ eˣ(cosx - sinx).             eˣ cosx

18) ∫ eˣ(cosx +  sinx).             eˣsinx

19) ∫ eˣ {(1+ sinx)/(1+ cos x).      eˣ tan(x/2)

20) ∫ eˣ {(2+ sin 2x)/(1+ cos 2x).     eˣ tan x

21) ∫ eˣ {(1- sinx)/(1- cos x).        - eˣ cot(x/2)

22) ∫ eˣ(cot x - cosec²x).        eˣ cot x

23) ∫ eˣ secx(1+ tanx).                   eˣ secx 

24) ∫ eˣ (tanx - log cosx).      eˣ log secx

25) ∫ eˣ (secx + log (secx + tanx)).     eˣ log (secx+ tanx)

26) ∫ eˣ (cotx + log sinx).       eˣ log sinx

26) ∫ eˣ (tanx + log secx).      eˣ log secx

27)  ∫ eˣ {(sin 4x - 4)/(1- cos 4x).       eˣ cot 2x

28) ∫ e²ˣ (- sin x + 2cos x).           e²ˣ cosx

29) ∫ eˣ{(sinx cosx -1)/sin²x).           eˣ cot x

30) ∫ eˣ{(sinx cosx +1)/cos²x).                eˣ tan x

31)  ∫ e⁻ˣ⁾² {(1- sinx)/(1+ cosx)}.       e⁻ˣ⁾² sec(x/2)

32) ∫ e²ˣ{(1+ sin 2x)/(1+ cos 2x)}.         e²ˣ tanx

33) ∫ eˣ{√(1- x²) sin⁻¹x +1}/√(1- x²).               eˣ sin⁻¹x

34) ∫{sin x (logx) + cos(logx)}.        x sin(log x)

35) ∫ {tan(logx) + sec²(logx)}.       x tan(log x)

36) ∫ eˣ{1/(1+ x²)  + tan⁻¹x}.          eˣ tan⁻¹x


  

             EXERCISE - 3
              -----------------

1) ∫ e²ˣ sin 3x.            e²ˣ/13 (2 sin 3x - 3 cos 3x)

2) ∫ eᵃˣ cos bx.      eᵃˣ/(a²+b²) (a cos bx + b sin bx)  

3) ∫ e⁻ˣ cosx.                  e⁻ˣ/2(sinx - cosx)

4) ∫ e⁻²ˣ sinx.          e⁻²ˣ/5(- 2 sinx - cosx)

5) ∫ sin(log x).            x/2 [sin(logx) - cos(log x)

6) ∫ cos(log x).          x/2 [cos(logx) + sin(log x)

7) ∫ eᵃˣ cos(bx +c).         eᵃˣ/(a²+ b²) {a cos(bx +c) + b sin(bx +c)

8) ∫ eᵃˣ sin(bx +c).           eᵃˣ/(a²+ b²) {a sin(bx +c) - b cos(bx +c)}

9) ∫ e²ˣ cos(3x +4).            e²ˣ/13  {2 cos(3x +4) + 3 sin(3x +4)}

10) ∫ eˣ cos²x.                         eˣ/2 +  eˣ/10( cos2x + 2 sinx)

11) ∫ e²ˣ cos²x.                       e²ˣ/4 +  e²ˣ/8 ( cos2x + 2 sin2x)

12) ∫ eˣ sin²x.                  eˣ/2 - eˣ/10 ( cos2x + 2 sin 2x)

13) ∫ 1/x³ sin(logx)              -1/5x² [cos(log x) + 2sin(logx)]

14) ∫ e²ˣ sinx cos x.             e²ˣ (sin 2x  - cos 2x)

15) ∫ e²ˣ sinx.       e²ˣ/5 (2sinx - cos x)

16)  ∫ x² ₑx³ cos x³.        ₑx³/6 (sin x³ + cos x³)





INTEGRAL OF THE FORM 
∫√(ax²+ bx+ c) dx

For Solution: 
STEP -1: Make coefficient of x² as one by taking 'a' common to obt6 x² + bx/a + c/a 

STEP -2:  Add and subtract (b/2a)² in x² + bx/a + c/a to obtain (x + b/2a)² + (4ac - b²)/4a².
After applying these two steps the integral reduces to one of the following three forms:
* ∫√(a²+x²) dx.
* ∫√(a²- x²) dx.
* ∫√(x² - a²) dx 

STEP-3: Use the appropriate formula.

1) ∫√(a²+x²) dx = x/2 √(a² + x²) + a²/2 log|x + √(a² + x²)|

2) ∫√(a²- x²) dx= x/2 √(a² - x²) + a²/2 sin⁻¹(x/a)

3) ∫√(x² - a²) dx = x/2 √(x² -a²) - a²/2 log|x + √(x² - a²)|



              EXERCISE- 4
                --------------

1) ∫√(x²+ 2x +5) dx.           1/2 (x+1) √(x²+ 2x +5) + 2 log|(x+1) + √(x²+ 2x +5)|

2) ∫√(x²+ x +1) dx.              (2x+1)/4 √(x²+ x +1) + 3/8 log|(2x+1) + √(x²+ x +1)|

3) ∫√{(x-3)(5- x)} dx.           1/2 (x -4) √{(x -3)(5-x) + 1/2 sin⁻¹{(x-4)/1}

4) ∫√(7x - 10 - x²) dx.         1/4  (2x-7) √(7x -10 -x²) + 9/8 sin⁻¹{(2x-7)/3}

5) ∫√(3+ 2x - x²) dx.         1/2 (x-1) √(3+ 2x -x²) + 2 sin⁻¹{(x- 1)/2}

6) ∫√(1+ x - 2x²) dx.          1/8 (4x-1) √(1+ x - 2x²) + 9√2/32 sin⁻¹{(4x- 1)/3}

7) ∫√(2x² + 3x+4) dx.            1/8 (4x+3) √(2x² +3x+4) + 23√2/32 log|{(x +3/4) + √(x² +3x/2 + 2)}

8) ∫√(3- 2x - 2x²) dx.         1/4 (2x+1) √(3- 2x - 2x²) + 7/4√2  sin⁻¹{(2x+ 1)/√7}

9) ∫√(4x² +9) dx.         x/2 √(4x² +9) + 9/4 log|2x + √(4x²+9)|

10) ∫√(x -x²) dx.         1/4 (2x -1) √(x -x²) + 1/8 sin⁻¹{(2x-1)}

11) ∫√(9 -x²) dx.         x/2 √(9 -x²) + 9/2 sin⁻¹{x/3}

12) ∫√(3 -x²) dx.         x/2 √(3 -x²) + 3/2 sin⁻¹{x/√3}

13) ∫√(2ax -x²) dx.         1/2 (x -a) √(2ax -x²) + a²/2 sin⁻¹{(x -a)/a}

14) ∫√(16x² +25) dx.         2x √(x² + 25/16) + 25/8 log|x+ √(x² + 25/16)|

15) ∫√(4x² - 5) dx.         x √(x² - 5/4) - 5/4 log|x + √(x² - 5/4)|

16) ∫ x√(x⁴ +1) dx.          1/4  x²√(x⁴+ 1) + 1/4 log|x² + √(x⁴ + 1)|

17)  ∫ x²√(a⁶ - x⁶) dx.          1/4  x³√(a⁶ - x⁶) + a⁶/6 sin⁻¹{x³/a³}

18) ∫ √{16+ (log x)²}/x.        1/2 log x √{(logx)²+ 16} + 8 log|log x + √{(logx)² +16}|

19) ∫ eˣ √(e²ˣ +1) dx.         eˣ/2 √(e²ˣ +1) + 1/2 log|eˣ + √(e²ˣ +1)|

20) ∫ cos x √(4 - sin²x) dx.          1/2 sin x √(4 - sin²x) + 2 sin⁻¹{1/2 sinx}

21) ∫√{(1- √x)/(1+ √x)} dx.         (1- x) (√x -2) - sin⁻¹√x




INTEGRALS OF THE FORM 
∫ (px +q) √(ax² + bx + c) dx.

For Solution:

STEP-1: Express px + q as
px + q= K d/dx (ax² + bx + c) + M
i e. px +q = K(2ax+ b)+ M

STEP-2: Obtain the values of K and M by equating the coefficient of x and constant terms on both sides.

STEP-3: Replace px +q by K(2ax +b) + M in the integral to obtain.

∫ (px+q) √(ax² + bx +c) dx =
     K ∫ (2ax+ b) √(ax²+bx +c) dx + M ∫ √(ax² + bx +c) dx.


STEP-4: To evaluate first integral on RHS, use the formula 
∫ (f(x))ⁿ f'(x) dx = (f(x))ⁿ⁺¹/(n+1).
      Evaluate second integral on RHS by the method as Exercise-4


              EXERCISE -5
               -----------------

1) ∫ (x -5) √(x²+ x) dx.                      1/3 √(x²+x)³ - 11/7 [(2x+1)/4 √(x²+x) - 1/8 log|(x + 1/2) + √(x²+x)|]

2) ∫ (3x -2) √(x²+ x +1) dx.         √(x²+x+1)³ - 7/2 [(x+1/2) √(x²+x +1) + 3/8 log|(x + 1/2) + √(x²+x+1)|]

3) ∫ (x +1) √(x²- x +1) dx.                      1/3 √(x²-x +1)³ + 3/8 {(2x-1) √(x²-x+1) + 9/16 log|(x - 1/2) + √(x²+x+1)|

4) ∫ (x +1) √(2x²+ 3) dx.                      1/6 √(2x²+3)³ + x/2 √(x²+3)+ 3√2/4 log|{x √2 + √(2x²+3)}/√3|

5) ∫ (x +2) √(x²+ x+1) dx.                      1/3 √(x²+x+1)³ + 3/8(2x+1) √(x²+x+1) + 9/16 log|(x + 1/2) + √(x²+x +1)|

6) ∫ (4x +1) √(x²- x-2) dx.                      4/3 √(x²-x-2)³ + 3/4 (2x-1) √(x²-x-2) - 27/8 log|(x - 1/2) + √(x²-x-2)|

7) ∫ (x -2) √(2x²-6 x+5) dx.                      1/6 √(2x²-6x+3)³ - 1/√2 [(2x-3)/4 √(x²-3x+ 5/2) + 1/8 log|(2x -3)/2 + √(x²-3x+ 5/2)|]

8) ∫ (x +1) √(x²+ x+1) dx.                      1/3 √(x²+x+1)³ +1/3 [(2x+1)/4 √(x²+x +1) + 3/8 log|(x + 1/2) + √(x²+x +1)|]

9) ∫ (2x +3) √(x²+ 4x+3) dx.                      2/3 √(x²+4x+3)³ -  [(x+2)/2 √(x²+4x+3) - 1/2 log|(x + 2) + √(x²+4x+3)|]

10) ∫ (2x -5) √(x²- 4x+3) dx.                      2/3 √(x²-4x+3)³ - [(x-2)/2  √(x²-4x+3) - 1/2 log|(x - 2) + √(x²- 4x +3)|]

11) ∫ x  √(1- x²+ x) dx.                     - 1/3 √(1 - x²+x)³ + 1/2 [(x- 1/2)/2 √(1-x²+x) + 5/8 sin⁻¹{(2x-1)/√5}

12) ∫ (x+1) √(1- x²- x) dx.                      - 1/3 √(1 - x²-x)³ + 1/8 (2x+ 1/2) √(1-x²-x) + 5/16 sin⁻¹{(2x+1)/√5}

13) ∫ (2x-5)  √(2- x²+ 3x) dx.                     - 2/3 √(2 - x²+3x)³ - (2x- 3)/2 √(2-x²+3x) - 17/4 sin⁻¹{(2x-3)/√17}

14) ∫ x √(x²+ x) dx.                     1/3 √( x²+x)³ - 1/8 (2x +1) √(x²+x) + 1/16 log |(x + 1/2) + √x² +x)|














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Wednesday, 5 January 2022

EQUATION OF STRAIGHT LINE (XI)


                  EXERCISE--1
                  -------------------..


1) Find the slope of a line whose inclination is:
a) 30°.                                       1/√3
b) 120°.                                       - √3
c) 135°.                                          -1
d) 90°.                             not defined
e) -π/4.                                           -1
f) 2π/3.                                        - √3
g) 3π/4.                                          -1
h) π/3.                                            √3

2) Find the inclination of a line whose slope is
a) √3.                                             60°
b) 1/√3.                                         30°
c) 1.                                               45°  
d) -1.                                            135°
e) -√3.                                          120°

3) Find the slope of a line which passes through the points:
a) (0,0) and (4, -2).                    -1/2
b) (0,-3) and (2, 1).                        2
c) (2,5) and (-4, -4).                     3/2
d) (-2,3) and (4, -6).                    -3/2

4) Show that the following points are collinear:
a) (1,5),(3,14),(-1,-4).
b) (3,-4),(1,2),(2,1)
c) (3,-2),(-1,1),(-5,4)
d) (4,7),(-2,-5),(2,3)


5) If the slope of the line joining the points A(x,2) and B (6, -8) is -5/4, find the value of x.                         -2

6) Find y if the slope of the line joining (-8,11),(2,y) is -4/3.        -7/3

7) Show that the line through the points (5,6) and (2,3) is parallel to the line through the points (9,-2) and (6,-5).     

8) Find the value of x so that the line through (3,x) and (2,7) is parallel to the line through (-1,4) and (0,6).                                          9 

9) Show that the line through the point (-2,6) and (4,8) is perpendicular to the line through the points (3,-3) and (5,-9).   

10) If A(2,-5), B(-2,5), C(x,3) and D(1,1) be four points such that a AB and CD are perpendicular to each other, find the value of x.                6

11) Without using Pythagora's theorem, show that the points
a) A(1,2), B(4,5) and C(6,3) 
b) A(0,4), B(1,2) and C(3,3) are the vertices of a right-angled triangle.

12) Using slopes, find the value of x for which the points A(5,1), B(1,-1) and C(x,4) are collinear.                11

13) Find the value of x for which the points (x, -1),(2,1) and (4,5) are collinear.                                           1

14) Using slopes, show that point
a) A(-4,-1), B(-2,-4), C(4,0) and D(2,3)
b) (-4,-1),(-2,-4),(4,0),(2,3) 
 are taken in order, are the vertices of a rectangle.  

15) Using slopes prove that the points A(-2,-1), B(1,0), C(4,3) and D(1,2) are the vertices of a parallelogram.

16) If the points A(a,0), B(0,b) and P(x,y) are collinear, using slopes, prove that x/a + y/b = 1.

17) If three A(h,0), P(a,b) and B(0,k) lie on a line, show that: a/h + b/k = 1.

18) A line passes through the points A(4,-6) and B(-2,-5). Show that the line AB makes an obtuse angle with the x-axis.

19) The vertices of a quadrilateral are A(-4,2), B(2,6), C(8,5) and D(9,-7). Using slopes, show that the midpoints of the sides of the quadrilateral ABCD form a parallelogram.

20) Find the slope of the line which makes an angle 30° with the positive direction of the y-axis, measured anticlockwise.             -√3

21) Find the angle between the lines whose slopes are
a) √3 and 1/√3.                           30°
b) 2 and -1.       
c) 2 and 3/4
d) (2- √3) and (2+ √3).               60°

22) Find the slope of the line which makes an angle of 45° with a line of slope -6/5.                           11, -1/11

23) Find the interior angles of the triangle whose vertices are A(4,3), B(-2,2) and C (2,-8).   

24) If A(1,2), B(-3,2) and C(3,-2) be the vertices of a ∆ABC, show that 
a) tan A= 2.
b) tan B= 2/3
c) tan C= 4/7

25) If t is the angle between lines joining the points A(0,0) and B(2,3), and the points C(2,-2) and D(3,5), Show that tan t= 11/23.

26) If t is the angle between the diagonals of a parallelogram ABCD whose vertices are A(0,2), B(2, -1), C(4,0) and D(2,3). Show that tan t = 2.

27) Show that the points A(0,6), B(2,1) and C(7,3) are three corners of a square ABCD. find
a) the slope of the diagonal BD.  7/3
b) the co-ordinates of the fourth vertex D.                                       (5,8) 

28) A(1,1), B(7,3) and C(3,6) are the vertices of a ∆ABC. If D is the midpoint of the BC and AL perpendicular to BC. find the Slope of 
A) AD.                                            7/8
B) AL.                                            4/3

29) The slope of a line is double of the slope of another line. If tangents of the angle between them is 1/3, find the slope of the other line.                                  1, 1/2

30) A quadrilateral has vertices (4,1),(1,7),(-6,0) and (-1,-9). Show that the midpoints of the sides of this quadrilateral form a parallelogram.

               EXERCISE -2
                 ----------------

1) Find the equation of a line parallel to the x-axis at a distance of 
a) 4 units above it.                  y-4= 0
b) 5 units below it.                  y+5= 0

2) Find the equation of a line parallel to the y-axis at a distance of 
a) 6 units to its right.             x- 6= 0
b) 3 units to the left.             x+ 3= 0

3)a) Find the equation of a line parallel to the x-axis and having intercept -3 on y-axis.              y+3= 0

b) Find the equation of the line parallel to x-axis and having intercept -2 on y-axis.                y= -2

c) Find the equation of the line parallel to x-axis and passing through (3,-5).                              y=-5

d) Find the equation of the line perpendicular to x-axis and having intercept -2 on x-axis.                x= -2


4) Find the equation of a horizontal line passing through the point (4,-2).                                        y+2 = 0

5)a) Find the equation of a vertical line passing through the point (-5, 6).                                              x+5= 0

b) Find the equation of the straight lines which pass through (4,3) and are respectively parallel and perpendicular to the x-axis.   y=3, x=4

6)a) Find the equation of a line which is equidistant from the lines x= -2 and x= 6.                           x= 2

b) Find the equation of a line equidistant from the lines y= 10 and y= -2.                                             y= 4

7) Find the equation of a line which is equidistant from the line y= 8 and y= -2.                                            y= 3

8) Find equation of a line
a) whose slope is 4 and which passes through the point (5,-7).    4x - y- 27= 0 
b) Whose slope is -3 and which passes through the point (-2,3). 3x+ y+3= 0
c) Which makes an angle of 2π/3 with the positive direction of the x-axis and passes through the point (0,2).                             √3 x+ y -2=0

9) Find the equation of a line whose inclination with the x-axis is 30° and which passes through the point (0,5).                       x - √3 y + 5√3 = 0

10) Find the equation of a line whose inclination with the x-axis is 150° and which passes through the point (3,-5).     x+ √3 y +(-3+5 √3)= 0

11) find the equation of a line passing through the origin and making an angle 120° with the positive direction of x-axis.        √3 x + y= 0

12) find the equation of a line which cuts off intercept 5 on the x-axis and makes an angle 60° with the positive direction of x-axis.      √3 x - y -5√3= 0

13) Find the equation of the line passing through the point P(4,-5) and parallel to the line joining the points A(3,7) and B (-2,4).

14) find the equation of the line passing through the point P(-3,5) and perpendicular to the line passing through the points A(2,5) and B (-3,6).                  5x - y+20= 0 

15) find the slope and the equation of the line passing through the points
A) (3,-2) and (-5,-7).              5/8, 5x- 8y-31= 0
B) (-1,1) and (2,-4).               -5/3, 5x+ 3y+2= 0
C) (5,3) and (-5,-3).          3/5, 3x- 5y= 0
D) (a,b) and (-a,b).                   0, y= b

16) Find the angle which the line joining the points (1,√3) and (√2, √6) makes with the x-axis.           60°

17) Prove that the points A(1,4),.B(3,-2) and C (4,5) are collinear. also find the equation of the line on which these points lie.          3x+y= 7

18) If A(0, 0) B(2,4) and C(6,4) are the vertices of a ∆ABC, find the equations of its sides.         y= 4, 2x-3y= 0, 2x -y= 0.

19) If A(-1,6), B(-3,-9) and C(5,-8) are the vertices of a ∆ ABC, find the equation of its medians.     29x+4y+5= 0, 8x- 5y-21, 13x+ 14y+ 47= 0.

20) Find the equation of the perpendicular bisector of the line segment whose end points are A(10,4) and B(-4,9).          28x - 10y -19= 0
 
21) Find the equations of the altitudes of a ∆ABC, Whose vertices are A(2,-2), B(1,1) and C(-1,0).       2x+y -2= 0, 3x- 2y- 1= 0, x- 3y -1= 0

22) If A(4,3), B(0,0) and C (2,3) are the vertices of a ∆ABC, find the equation of the bisector of angle A. x- 3y+5= 0

23) The midpoints of the sides BC, CA and AB of a ∆ ABC are D(2,1), E(-5,7) and F(-5,-5) respectively. Find the equations of the sides of if 1423 are the third season of ∆ABC.          x- 2= 0, 6x - 7y +79= 0, 6x+ 7y +65= 0

24) If A(1,4), B(2,-3) and C(-1,-2) are the vertices of a∆ ABC, find the equation of
a) the median through A..       13x- y -9= 0
b) the altitude through A.          3x-y+1= 0
c) the perpendicular bisector of BC.       3x- y - 4= 0

             
                  EXERCISE -3
                    ---------------


1)a) Find the equation of a line making an angle of 150° with the x-axis and cutting off an intercept 2 from y-axis.                  x +√3 y = 2√3

b) Find the equation of the line which makes an angle of 30° with the positive direction of the x-axis and cuts off an intercept of 4 units with the negative direction of the y-axis.                           x - √3y - 4√3= 0

c) Find the equation of the line whose inclination is 5π/6 and which makes an ntercept of 6 units on the negative direction of the y-axis.                               x + √3y + 6√3= 0


2) Find the equation of a straight line :
a) with slope 2 and y-intercept 3.     y= 2x+3
b) with slope -1/3 and y-intercept - 4.             x+ 3y+12= 0
c) with slop -2 and intersecting x- axis at a distance of 3 units to the left of the origin.    2x+ y+6= 0
d) Slope= 3, y-intercept=5.         3x-y+5= 0
e) Slope=-1, y-intercept= 4.     x+y-4= 0
f) Slope= -2/5, y-intercept= -3.     2x+ 5y+15= 0


3) Find the equation of a line which makes an angle of tan⁻¹(3) with the x-axis and cuts off intercept of 4 units on negative direction of y-axis.                                     y= 3x -4

4) Find the equation of the line cutting off an intercept -2 from the y-axis and inclined to the axes.     x- y - 2= 0, or x+ y +2= 0

5) Find the equation of a line that has y-intercept -4 and is parallel to the line joining (2,-5) and (1,2).     7x + y+4= 0

6) Find the equation of the line through the point (-1,5) and making an intercept of -2 on the y-axis.  7x+ y + 2= 0

7) find the equation of a line which is perpendicular to the line joining (4,2) and (3,5) and cuts off an intercept of length 3 on y- axis.       x - 3y+9= 0

8) find the equation of the perpendicular to the line segment joining (4,3) and (-1,1) if cuts off an intercept -3 from y-axis.     5x+ 2y+ 6= 0

9) find the equation of the straight line intersecting y-axis at a distance of 2 units above the origin and making an angle 30° with the positive direction of the x-axis.     x - √3 y+ 2√3= 0

10) Find the equation of the line which is parallel to the line 2x - 3y = 8 and whose y-intercept is 5 units.       2x- 3y +15= 0

11) Find the equation of the line which is perpendicular to the line x - 2y = -5 and passing through (0,3).       2x + y -3= 0     

12) Find the equation of the line passing through the point (2,3) and perpendicular to the line 4x +3y= 10.                         3x- 4y +6= 0

13) Find the equation of the line passing through the point (2,4) and perpendicular to the x-axis.      x = 2

14) Find the equation of the line that has x-intercept -3 and which is perpendicular to the line 3x +5y= 4.       5x- 3y + 15= 0

15) Find the equation of the line passing through the midpoint point of the line joining the point (6,4) and (4, -2) and perpendicular to the line 3x +2y= 8.              2x- 3y -7= 0

16) Find the equation of the line whose y-intercept is -3 and which is perpendicular to the line joining the points (-2,3) and (4,-5).             3x- 4y -12= 0

17) Find the equation of the line passing through (-3,5) and perpendicular to the line through the points (2,5) and (-3,6).    5x - y+20= 0

18) A line perpendicular to the line segment joining the points (1,0) and (2,3) divides it in the ratio 1:2. Find the equation of the line.    3x+ 9y-13= 0


MISCELLANEOUS-1 (A)
-------------------------------

1) Find the gradient of the line joining the pair of points:
a) (0,3),(4,5).                                1/2
b) (0,-2),(-2,4).                                -3
c) (1/2,3/2),(5/2,7/2).                    1
d) (√3+1,2),(√3+3,4).                      1

2) Find the Slope of a line through each of the pair of the following points:
a) (2,3) and (3,4).                             1
b) (2,-1) and (4,1).                            1
c) (-3,-2) and (-2,-1).                         1
d) (-5,-3) and (4,3).                       2/3

3) Find the inclination of the line joining the pair of points:
a) (1,2),(2,3).                                 45°
b) (-1,-3),(3,1).                               45°


4) If A(4,-3), B(6,5) and C(5,1) are three points, find the slope of AB and BC. Hence show that the points are collinear.                                 4, 4

5) Show that the following points are collinear:
a) A(5,-2), B(4,-1) and C(1,2). Hence find the inclination of the line AC.             135°

6)a) If (-5,a),(3,6) and (7,8) are collinear, find a.                                2

b) If A(2,a), B(3,-1) and C(4, -5) are collinear, find a.                               3

c) Find a, if the points (-2,3),(3,4),(a,5) are collinear.                            8

d) If the points (a,1),(1,2) and (0, b+1) are collinear, Show that 1/a + 1/b = 1.

7)a) If 2y - p²x= 3 and 2y - 4px +1= 0 are parallel, find the value of p.       4

b) If y - 2x = 3 and 2y = px +8 are parallel, find the value of p.             4

c) If 3(k -1)y - 6x = 2 and 4y - 8x +10= 0 are parallel, find the value of k.                                                       2

8)a) If (p+1)x + y= 3, and 3y - (p- 1)x= 4 are perpendicular, Find the value of p.                                       ±2

b) If 2my - 3x= 4 and 3my + 8x =10 are perpendicular to each other, find m.                                                      ±2

c) If y+ (2p+1)x +3= 0 and 8y- (2p -1)x= 5 are perpendicular, find the value of p.                                   ±3/2

9) State the slope (m) and y-intercept (c) of the line 2y = 4x - 3.             2, -3/3

10) Find the slope of a line
a) parallel 
b) perpendicular to to 2y= 3x+1.
     3/2, -2/3


MISCELLANEOUS-- 1(B)
-----------------------------

1) State the slope(m) and y-intercept(c) of the line:
a) y= x+1.                                       1,1
b) 2y= 4x+1.                                2,1/2
c) 3y= 6x -2.                              2, -2/3
d) 2y= - 3x -4.                            -3/1, 2

2) Find the Slope of a line which is parallel to:
a) 2y= x+2.                                     1/2,
b) 3y/4= 2x -2.                               8/3


3) Find the Slope of a line which is perpendicular to:
a) 3y= x+5.                                       -3
b) 2y/3= x -3.                              -2/3

4)a) Find the equation of the line y-intercept of 3, and a slope of 2.    y= 2x+ 3.      

b) Find the equation of the line y-intercept of 7, and a slope of 2.    y= 2x+ 7.      

c) Find the equation of the line with y-intercept of 4 and a slope of -3.        y= - 3x +4.      

d) Find the equation of the line with y-intercept of 1/2 and a slope of 2.       2y= 4x+1.     

e) Find the equation of the line with y-intercept of 5, and a slope of -2/3.     3 y= - 2x+15.      

f) Given, y-intercept of 2, and an inclination of 45° with the positive direction of the x-axis.        y= x+ 2.      

g) Find the equation of the line having y-intercept of 4 and which is equally inclined to the axes, in the second quadrant.                  y= x+ 4

5)a) Find the equation of a line through (0,3) and having slope = 4.     y= 4x +3

b) Find the equation of a line through (1,2) and having slope = 3.     y= 3x - 1


6)a) Find the equation of the line through (0,2) and parallel to y= 3x +2.                    y= - x+ 3.      

b) Find the equation of the line through (4,0) and parallel to 3y= 6x +2.                                         y= 2x - 8     
c) Find the equation of the line through (0,3) and parallel to 2y= x - 2.                                              2y= x -6

7) Find the equation of the line through (0,2) and perpendicular to
a) y= 2x+3.                    2y= - x+4.     
 
b) y= x/3 +2.                        y= - 3x+2

c) Find the equation of the line through (0,3) and perpendicular to 2y= x + 1.                      y= - 2x+ 3.      


8)a) Find the equation of the line through (0,2) and (-2,0).        y= x+2

b) Find the equation of the line through (3,0) and (0,3).         y= -x+3

c) Find the equation of the line through (2,1) and (4,3).         y= x -1

9)a) Find the equation of the line through (2,3) and (3,4) and y-intercept of 5 units.                y= x+ 5

b) Find the slope of the line joining the points (3,4) and (0,16). Hence or otherwise, write down the equation of this line.                  -4, y= - 4x+16

10) The equation of a line is y= 3x -5. Write down the slope of this line and intercept made by it on the y-axis. Hence or otherwise, write down the equation of a line which is parallel to this line and which passes through the point (0,5).      3, -5, y= 3x+5 

11) A(2,1), B(5,3), C(-1,3) are the vertices of the triangle ABC. Find
a) equation of the median AD. 2= x
b) the equation of the altitude BE.   2y= 3x -9     
c) the equation of the altitude CF.   2y= -3 x+ 3      

12) find the equation of the line through (-4,8) and parallel to x-axis.        y= 8

13) find the equation of the line through (3,5) and perpendicular to the axis.                                         y= 5

14) A(1,3) and C(6,8) are the opposite vertices of a square ABCD. Find the equation of the diagonal BD.                                     x +y = 9

15) Find the equation of the perpendicular bisector of the line segment joining (3,2) and (7,6).     x+ y = 9

16) A(-1,4) and B(5, -2) are two points. Find the equation of the perpendicular bisector of AB.     y + x -3= 0 

17) A straight line cuts off, on the axes of co-ordinates positive intercepts whose sum is 7. If the line passes through the point (-3,8), find its equation.          4x +3y= 12

18) A straight line cuts off, on the axes, positive intercepts whose sum is 7. If the line passes through the point (-8, 9), find its equation.    3x +4y= 12

19) The coordinates of the vertex A of a square ABCD are (1,2) and the equation of the diagonal BD is x + 2y= 10. Find the equation of the diagonal and the coordinates of the centre of the square.      y- 2x= 0; (2,4)

20) A, B are the points (0,6) and (10,0). O is origin, OM is a median and OP an altitude of triangle AOB. Find the equation of OM and OP.          5y- 3x= 0; 3y- 5x= 0

21) Find the equation of the line passing through the point (3,4) such that the portion between the axes is divided by P in the ratio 2:3.       y + 2x= 10

22) Find the equation of a line, which has the y-intercept of 5 and is parallel to the line 4x - 6y= 9. Find the coordinates of the point, where it cuts the x axis.              3y- 2x= 15, (-7.5,0)



              EXERCISE -4
               ----------------

1) Find the equation of the line which cuts off intercepts 
a) - 3 and 5                 5x - 3y +15= 0
b) -2, and 3.                3x - 2y +6= 0
c) -k/m and k.             mx - y + k= 0
d) 4 and -6 
on the x- axis and y- axis respectively.                3x - 2y -12= 0

2) Determine the x-intercept and y-intercept of the following:
a) 3x+ 5y -15= 0.                         5, 3
b) x - y - 7= 0.                             7, -7


3) Find the equation of the line that cuts off equal intercepts on the co-ordinate axes and passes through the point 
a) (4,7).                            x + y -11= 0
b) (2,3).           x+y-5= 0 or x- y+ 1= 0

4)a) Find the equation of the line which passes through the point (3,-5) and cut off intercept on the axes which are equal in magnitude but opposite in sign.        x - y -8= 0

b) Find the equation of the line which passes through the point (5, 6) and cut off intercept on the axes which are equal in magnitude but opposite in sign.        x - y +1 = 0


5) Find the equation of the line which makes an intercept of 2a on the x-axis and 3a on the y-axis. Given that the line passes through the point (14,9), find the numerical value of a.             3x+ 3y- 6a= 0, 4

6) Find the equation of the line passing through the point (2,2) and cutting off intercepts on the axes, whose sum is 9.     x + 2y -6= 0 or 2x +y -6 = 0

7) A straight line passing through (2,3) and the portion of the line intercepted between the axes is bisected at this point. Find the equation.                  3x + 2y -12= 0

8) Show that the three points (5,1),(1,-1) and (11,4) lie on a straight line. Further find
a) its intercepts on the axes.     3
b) the length of the portion of the line intercepted between the axes.  -3/2
c) the slope of the line.               1/2

9) Find the equation of the line  passing through the point (22,-6) and whose intercept on the x-axis exceeds the intercept on the y-axis by 5.     6x + 11y -66= 0 or x +2y - 10 = 0

10) Find the equation of the line whose portion intercepted between the axes is bisected at the point (3, -2).       2x - 3y - 12= 0 

11)a)  Find the equation of the line whose portion intercepted between the coordinates axes is divided at the point (5,6) in the ratio 3:1.     2x + 5y - 40= 0 

b) Find the equation of the line whose portion intercepted between the coordinates axes is divided at the point (3, -2) in the ratio 4: 3     3x + 4y - 1= 0 

12) A straight line passes through the point (-5,2) and the portion of the line intercepted between the axes is divided at this point in the ratio 2:3. Find the equation of the line.          3x - 5y + 25= 0

13)a) If the straight line x/a + y/b = 1 passes through the points (8,-9) and (12,-15), find the value of a and b.             2, 3

b) If the straight line x/a - y/b = 1 passes through the points (8, 6) and cuts off a triangle of area 12 units from the axes of coordinates. Find the equation of the straight line.      x/4 - y/6 = 1 and x/8 - y/3 = - 1 


                  EXERCISE- 5
                 ---------------------

1) Find the equation of the line for which:
a) p= 3 and φ= 45°.        x+y -3√2= 0
b) p= 5 and φ= 135°.      x-y +5√2= 0
c) p= 8 and φ= 150°.   √3 x-y +16= 0
d) p= 3 and φ= 225°.     x+y +3√2= 0
e) p= 2 and φ= 300°.      x- √3y -4= 0
f) p= 4 and φ= 180°.        x+ 4= 0

2) The length of the perpendicular segment from the origin to a line is 2 units and the inclination of this perpendicular is φ such that sin φ= 1/3 and φ is acute. Find the equation of the line.         2√2 x + y - 6= 0

3) Find the equation of the line which is at a distance of 3 units from the origin such that tan φ= 5/12, where φ is the acute angle which this perpendicular makes with the positive direction of x-axis.    12 x + 5y - 39= 0


               EXERCISE-6
               -----------------

1) Reduce the equation to slope intercept form, and find from it the slop and y+intercept:
A) 2x - 3y - 5= 0.         y= 2x/3 - 5/3, 2/3 and -5/3
B) 5x + 7y - 35= 0.     y= -5x/7 + 5, -5/7 and 5   
C) y+ 5= 0.        y= 0.x - 5, 0 and -5
d) 


2) Reduce the equation to intercept form. Hence, find the length of the portion of the line intercepted between the axes.
A) 3x - 4y+12= 0.       x/-4 + y/3 = 1, 5 units
B) 5x - 12y= 60.         x/12 + y/-5= 1, 13 units.

3) Find the inclination of the line:
a) x + √3 y +6= 0.                        150°
b) 3x + 3 y +8= 0.                       135°
c) √3 x - y - 4= 0.                           60°

4)a) Reduce the equation x+ y - √2= 0 to the normal form x cos φ + y sin φ = p, and hence find the value of φ, p.             x cos 45°+ y sin 45°= 1, φ= 45°, p= 1

b)  Reduce the equation x+ √3 y - 4= 0 to the normal form x cos φ + y sin φ = p, and hence find the value of φ, p.             x cos 60°+ y sin 60°= 2, φ= 60°, p= 2

c) Reduce the following equation to the normal form and find p and φ in each case:
i) x+ √3 y - 4= 0.                        2, π/3
ii) x+ y + √2 = 0.                      1, 225°
iii) x - y + 2 √2= 0.                   2, 135°
iv) x - 4= 0.                                   3, 0
v) y - 2 = 0.                                 2, π/2

5) Reduce each of the following equations to the normal form:
a) x+ y -2 = 0.       x cos 45°+ y sin 45°= √2
b) x+ y + √2 = 0.     x cos 225°+ y sin 225°= 1
c) x+ 5 = 0.       x cos 180°+ y sin 180°= 5
d) 2y - 3 = 0.       x cos 90°+ y sin 90°= 3/2
e) 4x+ 3y - 9 = 0.       x cos φ + y sin φ= p, where cos φ=4/5, sin φ=3/5 and p= 9/5

6) Reduce the equation √3 x + y +2= 0 to
a) slope-intercept form and find slope and y-intercept.           -√3, - 2
b) intercept form and find intercept on the axes.                         -2/√3, - 2
c) The normal form and find p and φ.                                             1, 210°

7) Put the equation x/a + y/b = 1 to the slope intercept form and find the slope and y-intercept.    -b/a, b

8) The perpendicular distance of a line from the origin is 5 units and its slope is - 1. Find the equation of the line.                               x + y - 5 √2= 0

9) Reduce the lines 3x - 4y +4= 0 and 3x + 4y -5 = 0 to the normal form and hence find which line is nearer to the origin.     3x - 4y +4= 0

10) Show that the origin is equidistant from the lines 4x + 3y + 10= 0; 5x - 12y + 26= 0 and 7x + 24y = 50.                

11) Find the values of φ and p, if the equation x cos φ + y sin φ = p is the normal form of the line √3 x + y +2= 0.                                             210°, 1


                  EXERCISE --7
                   -----------------

1) Write down the slopes of the following lines:
a) 2x+ 3y +1= 0.                           -2/3

b) 7x - 5y +8= 0.                            7/5

c) -11x - 6y = 0.                        -11/6

d) xx₁ + yy₁= a².                       -x₁/y₁
e) 3x + 4y - 2(x + x₁) - 5(y + y₁) + 2= 0.               1

2) Find the value of k such that the line (k - 2)x + (k + 3)y - 5 = 0 is
a) parallel to the line 2x - y +7= 0.   -4/3
b) perpendicular to it.                7

3) prove that the lines 
a) 3x + 4y - 7= 0 and 28x - 21y + 50= 0 are mutually perpendicular.    
b) px + qy - r = 0 and - 4px - 4qy + 5r = 0 are parallel.   

4)  find the slope of the line which is perpendicular to the line 7x + 11y - 2 = 0.         11/7 

5) Show that (2,-1) and (1,1) are on opposite sides of 3x + 4y = 6.

6) The sides of a Triangles are given by the equations 3x + 4y = 10, 4x - 3y = 5 and 7x + y+10=0; show that the origin lies within the triangle.

7) find by calculation whether the points (13,8), (26, -4) lie in the same, adjacent, or opposite angles formed by the straight lines 5x + 6y - 112=0, and 10x + 11y - 217=0.       Opposite

      

            EXERCISE --8
            ---------------------

1) Find the point of intersection of the following pairs of lines:
a) 2x - y+3= 0, x + y - 5= 0.     (2/13, 13/3)
b) 3x - 5y+ 5= 0, 2x + 3y - 22= 0.     (5,4)
c) 2x - 3y -7 = 0, 3x - 4y - 13= 0.     (11, 5)
d) bx + ay = ab, ax + by = ab.       (2ab/(a+b), ab/(a+b))


2) Find the coordinates of the vertices of a triangle, the equations of whose sides are:
a) x + y- 4= 0, 2x - y +3= 0, x -3 y+2= 0.     (1/3, 11/3),(-7/5,1/5),(5/2,3/2)
b) 

3) Find area of the triangle formed by the lines
a) y= 0, x = 2, x +2 y = 3.                 0
b) x + y- 6= 0, x - 3y -2= 0, 5x -3 y+2= 0.                                      12 sq. units


4) Find the equations of the medians of a triangle, the equations of whose sides are: 3x + 2y +6= 0, 2x - 5y +4 = 0, x -3 y - 6 = 0.      41x - 112y- 70= 0, 16x - 59y -120 = 0, 25x -53 y+ 50 = 0.       

5) Prove that the lines y= √3 x +1, y = 4, y = - √3 x +2 form an equilateral triangle.

6) Classify the following pairs of the lines are coincident, parallel or intersecting:
a) 2x + y- 1 = 0, 3x + 2y +5 = 0,   intersecting
b) x - y = 0, 3x - 3y + 5= 0.   Parallel
c) 3x + 2y- 4= 0, 6x + 4y - 8= 0.   Coincident

7) Find the equation of the line joining the points (3,5) to the point of intersection of the lines 4x + y- 1= 0, 7x - 3y - 35= 0.    12x - y- 31= 0

8) Find the equation of the line passing through the point of intersection of the lines 4x - 7y- 3= 0 and 2x - 3y +1 = 0 that has equal intercepts on the axes.    x + y +13 = 0

9) Show that the area of the triangle formed by the lines y = m ₁x, y= m₂x and y= c is equal to c²/4 (√33 + √11), m₁ , m₂ are the roots of the equation of x² (√3 + 2)x + √3 - 1= 0.      x+ y+13= 0

10) If the straight line x/a + y/b = 1 passes through the point of intersection of the lines x+ y - 3= 0 and  2x - 3y -1= 0 and is parallel to x- y- 6= 0, find a and b.              1, -1

b) Find the ortho-centre of the triangle whose angular points are (0,0),(2, -1),(-1,3).                     (-4,-3)

11) a) Find the orthocentre of the triangle is equations of whose sides are x+ y -1 = 0,  2x+ 3y = 6 and 4x- y+4 = 0.          (19/7, 18/7)

12) Three sides AB, BC, CA of a triangle ABC are 5x - 3y+ 2= 0, x - 3 y - 2= 0 and x+ y - 6 = 0 respectively. Find the equation of the altitude through the vertex A.    3x+ y -10= 0

13) Find the coordinates of the orthocentre of the triangle whose vertices are (-1,3),(2,-1) and (0, 0).   (-4,-3)

14) Find the coordinates of the incentre and centroid of the triangle whose sides have the equations 3x - 4 y = 0,  5x+ 12y = 0 and y -15= 0.  (-1, 8),(-16/3, 15)

15) Prove that the lines √3 x+ y= 0, x+ √3y = 0, √3 x + y = 1 and x+ √3 y= 1 form a rhombus.

16) The vertices of a triangle are A(0,5), B(-1, -2) and C(11,7). Write down the equations of BC and the perpendicular from A to BC and hence find the coordinates of the foot of the perpendicular.     3x -4y- 5= 0, 4x + 3y - 15; = 0, (3,1)

17) Find the equation of the line passing through the point of intersection of the two lines x + 2y + 3= 0, 3x + 4y +7= 0 and parallel to the straight line y - x = 8.    x- y= 0


              EXERCISE--9
             ---------------------

1) Prove that the following sets of three lines are concurrent:
a) 15x -18y +1= 0, 12x +10y -3= 0, 6x + 66y -11= 0.  
b) 3x - 5y - 11= 0, 5x + 3y -7 = 0, x -3 2y= 0.                             
c) x/a + y/b = 1, x/b +y/a = 2 , x = y.
d) (b+c) x + ay+  2= 0, (c+a)x + by +1= 0, (a+b) x - c y+ 1= 0                                
2)a)  For what value of K are the three lines  2x - 5y +3 = 0, 5x - 9y + K = 0, x - 2 y+ 1= 0 concurrent ?      4

b) For what value of m are the three lines x - y +1 = 0, 2(x+1) = y, y = mx + 3 concurrent.                               3 

c) For what value of m are the three lines 3x - 4y -13 = 0, 8x - 11y-33 = 0 and ,  2x - 3y + m= 0 are concurrent.                                      -7

3) If the three lines ax² + a²y +1 = 0, bx + b²y + 2 = 0, cx + c² y+ 1= 0 are concurrent, show that atleast two of three constants a, b, c are equal.

4) If a, b, c are in AP., Prove that the lines ax + 2y +2 = 0, bx + 3y + 2 = 0, cx + 4 y+ 1= 0 are concurrent.

5) Prove that the lines 5x + 3y -7 = 0, 3x - 4y = 10, x + 2 y = 0 meets in a point.

6) Show that the lines lx + my +n = 0, mx + ny + l = 0, x + ly+ m= 0 are concurrent if l+ m + n = 0.

7) Show that the lines x - y -6 = 0, 4x - 3y -20 = 0, 6x +5 y+ 8= 0 are concurrent. Also , find their common point of intersection.   (2,-4)

                EXERCISE-10
              ----------------------
1) a) Find the equation of a line passing through the point (2,3) and parallel to the line 3x - 4y +5= 0.    3x - 4y + 6= 0

b) Find the equation of a line passing through the point (4,5) and  is
i) parallel.                       3x - 3y - 2= 0
ii) perpendicular to the line 3x - 2y +5= 0.                           2x + y -20= 0

c) Find the equation of a line passing through the point (4,3) and is parallel to the line 3x - 4y +5= 0.       3x - 4y = 0


2)a) Find the equation of line passing through (3,-2) and perpendicular to the line x - 3y +5= 0.                                     3x +y -7=0 

b) Find the equation of a line passing through the point (4,3) and perpendicular to the line 3x - 4y +5= 0.                                     4x + 3y =  25

3) find the equation of the perpendicular bisector of the line joining the point (1,3) and (3,1).        x=y  

4) find the equation of the altitude of a ∆ ABC whose vertices are A(1,4), B(-3,2) and C(-5,-3).       2x + 5y -12= 0,  6x + 7y +4= 0, 2x +y +13 = 0

5) find the equation of a line which is perpendicular to the line √3 x - y +5= 0 and which cuts off intercept of 4 units with the negative direction of y-axis.            x + √3y + 4√3=0

6)  find the equation of a line perpendicular to the line √3 x -y +5 = 0 and at a distance of 3 units from the origin.      x + √3 y ± 6 =0 

7) Find the equation of the straight line through the point (a,b) and perpendicular to the line lx + my + n = 0.                      m(x -a) = l(y - b)= 0

8) Find the equation of the straight line perpendicular to 2x - 3y = 5 and cutting off an intercept 1 on the positive direction of x-axis.    8x + 2y - 3= 0

9)a)  Find the equation of the straight line perpendicular to 5x - 2y = 8 and which passes through the midpoint of the line segment (2,3) and (4,5).                 2x + 5y - 26 = 0

b) Find the equation of a line passing through the point of intersection of the straight line -x + y +7= 0, 2x + y - 2= 0.         4x+ 3y= 0


10) Find the equation of the line which has y-intercept equal to 4/3 and is perpendicular to 3x - 4y + 11= 0.                           4x + 3y - 4= 0

11)  Find the equation of the right bisector of the line segment joining the points on (a, b) and (a₁, b₁).   2x(a₁ - a) + 2y ((b₁ - b)+ (a² + b²) - (a₁² + b₁²) = 0

12) Find the image of the point (2,1) with respect to the line mirror x + y -  5= 0.                                         (4,3)

13) If the image of the point (2,1) with respect to the line mirror be (5,2), find the equation of the mirror.                         3x + y - 12= 0  

14) Find the equation oto the straight line parallel to 3x - 4y +6= 0 and passing through the middle point of the join of points (2,3) and (4, -1).                          3x - 4y - 5= 0

15) Prove that the line 2x - 3y +1 = 0, x + y - 4= 0, 2x - 3y - 2= 0 and x + y - 4= 0 form a parallelogram.

16) find the equation of a line drawn perpendicular to the line x/y + y/6 = 1 through the point where it meets the y-axis.    3x - 3y + 18= 0

17) The perpendicular from the origin to the line y = mx + c meets it at the point (-1,2). Find the values of m and c.                           1/2, 5/2

18) Find the equation of the right bisector of the line segment joining the points (3,4) and (-1,2).   2x + y - 5= 0 

19) The line through (h, 3) and (4,1) intersects the line 7x - 9y - 19= 0 at a right angle. Find the value of h.    22/9

20) find the image of the point (3,8) with respect to the lines  x + 3y - 7= 0 assuming the line to be a plane mirror.                                       (-1,-4)

21) find the coordinates of the foot of the perpendicular from the point (-1,3) to the line 3x - 4y - 16= 0.     (68/25, -49/25)

22) Find the projection of the point (1,0) on the line joining points (-1,2) and (5,4).                           (1/5,12/5)

23) find the equation of the straight line which cuts off intercepts on x-axis twice that on y-axis and is at a unit distance from the origin.   x + 2y ± √5 = 0.

24) The equation of perpendicular bisectors of the sides AB and AC of a triangle ABC are x - y + 5= 0 and x + 2y = 0 respectively. If the point A is (1,-2), find the equation of the line BC.                       14x + 23y - 40= 0



              EXERCISE -- 11
                --------------------

1) Find the distance of the point:
a) (3, -5) from 3x - 4y = 27.         2/5
b) (-2, 3) from  12x - 5y = 13.          4
c) (-4, 3) from 4(x +5)= 3(y -6).     13/5
d) (2, 3) from y = 4.                         1
e) (0, 0) from h(x+ h) + k(y +k) = 0.          √(h² + k²)
f) (4, ) from the line joining the points (4,1) and (2,3).              1/√2


2) Find the length of the perpendicular from the origin to each of the following lines:
a) 7x + 24y= 50.                      2 units
b) 4x + 3y= 9.                       9/5 units
c) x= 4.                                     4 units
d) the two points (a cos k, a sin k) and (a cos m, a sin m).                  a cos{(k -m)/2}
e) (cos k, sin k) and (cos m, sin m).   Cos {(k - m)/2}

3) Prove that the product of the lengths of particular drawn from the points A(√(a² - b²),0) and B(- √(a² - b²), to the line x/a cos k +y/b sink = 1 is b².

3) a) Find the values of k for which the length of the perpendicular from the point (4,1) on the line 3x - 4y +k = 0 is 2 units.                            2, -18

b) If p is the length of the perpendicular from the origin to the line x/a + y/b = 1, then prove 1/p² = 1/a² + 1/b².

c) If p and p' be the perpendicular from the origin upon the straight lines x sec k + y cosec k = a and x cos k - y sin k = a cos 2k. Prove that 4p² + p'² = a².

d) If the length of the perpendicular from the point (1,1) to line ax - by + c= 0 be unity, show that 1/c + 1/a - 1/b = c/2ab.

e) Show that the product of perpendiculars on the line x/a cos k + y/b sin k=1 from the points (±√(a² - b²),0) is b²

4) Show that the length of perpendicular from the point (7,0) to the line 5x + 12y= 9 is double the length of perpendicular to it from the point (2,1).    

5) The points A(2, 3), B(4,-1) and C(-1,2) are the vertices of ∆ABC. find the length of perpendicular from C on AB and hence find the area of ∆ABC.                    7/√5, 7

6) a) What are the points on the x-axis whose perpendicular distance from the line x/3 + y/4 =1 is 4 units.    (8,0) and (-2,0)

b) What are the points on the y-axis whose perpendicular distance from the line 4x - 3y=12 is 3 units.    (0,1) and (0,-9)

c) What are the points on the x-axis whose perpendicular distance from the line x/a + y/b =1 is a units.    [a/b (b ± √(a²+ b²) ,0) 

7) Find all the points on the line x + y= 4 that lie at a unit distance from the line 4x + 3y= 10.    (3,1),(-7,11)

8) The perpendicular distance of a line from the origin is 5 units and its slope is -1. find the equation of the line.   x + y= - 5√2 or x + y= 5 √2

9) Find the distance of the point of intersection of the lines 2x + 3y= 21 and 3x - 4y +11=0 from the line  8x + 6y +5= 0.                                59/10

10) Find the length of the perpendicular from the point (4,-7) to the line joining the origin and the point of intersection of the lines  2x - 3y +14= 0 and 5x + 4y= 7.             1

11) Find the distance of the point (1,2) from the straight line with Slope 5 and passing through the point of intersection of x+ 2y = 5 and x - 3y = 7.                132/√650


                  EXERCISE--12
                   --------------------

1) Determine the distance between the following pair of parallel lines:

a) 4x - 3y= -5 and 4x - 3y +7= 0.  2/5

b) 8x + 15y= 36 & 8x + 15y= -32.    4 

c) y= mx +c and y= mx +d.                              |d - c|/√(1+ m²)

d) p(x +y)+ q= 0 and p(x +y) - r = 0.           |q + r|/√(2p) 

2) The equation of two sides of a  square are 5x - 152y= 65 & 5x - 12y + 26=0. Find the area of the square.                                          49 

3) a) Find the equation of two straight lines which are parallel to x + 7y +2= 0 & at units distance from the point (2,-1).       x + 7y + 6 ± 5√2=0 

b) Find the equation of straight lines are parallel to 3x - 4y - 5= 0 at a units distance from it.       3x - 4y=0 or 3x - 4y= 10 

4) Prove that the lines 2x + 3y= 19 & 2x + 3y +7=0 are equidistant from the line 2x + 3y= 6.            

5) Find the equation of the line midway between the parallel lines 9x + 6y= 7 & 3x + 2y= 6.        18x + 12y +11= 0

6) Prove that the line 12x - 5y= 3 is mid parallel to the lines 12x - 5y= 7 and 12x - 5y= 13.

7) A vertex of a square is at the origin and its one side lies along the line 3x - 4y= 10. find the area of the square.                             4 sq.units


          EXERCISE --13
           ---------------------

1) Show that the area of the parallelogram formed by the lines 2x - 3y + a=0,  3x - 2y - a = 0, 2x - 3y + 3a =0and 3x - 2y - 2a= 0 is  2a²/5 square units.

2) prove that the area of the parallelogram formed by the x cos k + y sin k = p, x cos k + y sin k = q, x cos m + y sin m = r and x cos m + y sin m = d is ±(p + q)(r - s) cosec(k - m).

3) Prove that the four straight lines x/a + y/b =1,  x/b + y/a  = 2, x/a +y/b  =2 and x/a + y/a - 2=0 form a rhombus. find its area.       a²b²/|b² - a²| 

4) Show that the four lines ax ± b ± c = 0 encloses a rhombus whose area is 2c²/ab.

5) Prove that area of the parallelogram formed by the lines a₁x + b₁y + c₁ =0,  a₁x + b₁y + d₁= 0, a₂x + b₂y + c₂ =0,  a₂x +  b₂y + d₂ = 0 is |{(d₁ - c₁)(d₂ - c₂)}/(a₁ b₂ - a₂ b₁)| sq. units. Deduce the condition for these lines to form a rhombus.   P₁ P₂/sin k, (P₁ P₂are the distance between the pairs of parallel lines and k is the angle between two adjacent sides, for rhombus P₁ = P₂)

6) Prove that the area of the parallelogram formed by the lines 3x - 4y + a=0, 3x - 4y + 3a = 0, 4x - 3y - a =0 and 4x - 3y - 2a= 0 is 2a²/7 units.

7)  show that the diagonals of the parallelogram whose sides are lx + my + n =0,  lx + my + n' = 0, mx + ly + n =0and mx + ly + n' = 0  include an angle π/2.     (Use P₁ = P₂)

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Monday, 3 January 2022

LOCUS

1) A point moves in the xy-plane in such a way that its distance from the point (4,0) is always equal to its distance from the y-axis. Find the equation to the locus of the moving point.                                  y²=8(x- 2)

2) A point moves in the xy-plane in such a way that its distance from the point (a,0) is always greater than the abscissa of the point by a. Find the equation to the locus of the moving point.                 y²= 4ax

3) Find the equation to the locus of a point if Its distance from the x-axis is double its distance from the point (1,1).    4x²+ 3y²- 8x - 8y+ 8= 0

4) The distance of a moving from x-axis is twice the distance of the point from the point (1,1). Find the equation to the locus of the point.         4x²+ 3y² - 8x - 8y +8= 0

5) A point moves in a plane such that its distance from the point (2,3) exceeds its distance from the y-axis by 2. Find the equation of the locus of the point.                      (y- 3)² = 8x.

6) Find the equation of the locus of a moving point which is always equidistant from the points (-1,1) and (3,-2).                         8x - 6y= 11.

7) Find the equation of the locus of a moving point which is always equidistant from the points (a+ b, a- b) and (a- b, a+b).                         y= x

8) Find the equation of the locus of a point which always forms an isosceles triangle with the points (5,2) and (-3,6).                 y= 2(x+1).

9) A and B are two fixed points whose coordinates are (2,1) and (3,2) respectively. A point P moves in such a way that PA= 2PB always. Find the locus of P.            3x² + 3y²- 20x - 14y+47= 0

10) The coordinates of two points A and B are (-5,3) and (2,4) respectively. Find the locus of P(x,y) such that PA:PB= 3:2.         5x²+ 5y² - 76x - 48y+ 44= 0

11) Find the locus of the point which moves such that its distance from the points (3,1) and (-2, 5) are always equal.            10x- 8y +19= 0

12) Find the locus of a point which moves so that it is always at a distance 4 from (1,-2).       x²+ y² - 2x + 4y -11= 0

13) Find the locus of a point which moves so that its distance from (4,0) is always twice its distance from (1,0).                           x²+ y²= 4

14) P is such a variable point that the sum of the squares of its distances from the points (-2,0) and (2,0) is always equal to 40. Find the equation of the locus of P.           x²+ y²= 16

15) Find the locus of a moving point so that its distance from (0,4) is two-thirds of its distance from (0,9).                                x²+ y²= 36

16) A points moves so that its distance from the point (a,0) exceeds the distances from axis of y by a. find the locus.             y²= 4ax

17) Find the locus of a point which moves so that its distance from the y-axis is double its distance from the point (2,2).           3x²+ 4y² - 16x - 16y +32= 0

18) Find the locus of a point which moves such that its distance from the point (4,0) is √2 times of its distance from the y-axis.      x² - y² + 8x - 16= 0

19) Find the locus of a point which moves so that its distance from the point (0,5) is two thirds of its distance from the x-axis.           9x²+ 5y² - 90y + 225= 0

20) The points A and B are (-4,0) and (-1,0) respectively. A point P moves in such a way that PA: PB=2:1. find the locus of P.          x²+y²= 4

21) Find the locus of a point which moves so that the sum of squares of its distances from the two points (3,0) and (-3,0) is 36.       x²+ y²= 9

22) Find the locus of a point which moves so that the sum of the squares of its distances from the points (3,0) and (-3,0) is always equal to 50.                      x²+ y²= 16

23) Find the locus of a point which moves so that the sum of its distances from the points (4,0) and (-4,0) is 10.               9x²+ 25y²= 225

24) Find the locus of a point which moves such that the difference of its distances from (2,5) and (6,5) is 2.                 3x²-y²- 24x+ 10y +20= 0

25) Find the locus of the centre of the circles passing through the point (c,0) and (- c ,0).                x= 0

26) A(2,-3) and B(4,0) are two points. Find the locus of a point P such that the area of the triangle PAB is always for units.      3x- 2y= 4

27) A line segment, 16 units in length, moves so that its ends are always on the positive coordinate axes. Find the equation of the locus of its mid-point.                 x²+y²= 64

28) If the coordinates of two vertices of a triangle be A(-3,0) and B(3,0) and angle ACB=90°, Find the locus of the centroid of the triangle.                              x²+ y²= 11

29) The coordinates of a moving point P are (a sec t, b tan t), where t is a variable parameter. Find the locus of the point P.     x²/a² - y²/b² = 1

30) The coordinates of a moving point P are {(t-1)/(t+1), (2t+1)/(t+1)}, where t is a variable parameter. Find the locus of the point P.                               x- 2y+3= 0

31) AB is a line of fixed length, 6 units, joining the points A(t,0) and B which lies on the positive y-axis. P is a point on AB distance 2 units from A. Express the coordinates of B and of P in terms of t. Find the locus of P as t varies.           x²+ 4y²= 16

32) Show that the locus of the point of intersection of the straight lines x sin t - y(cos t -1)= a sin t; x sin t - y(cos t + 1) = - a sin t is x² + y² = a².

33) A(1,2) and B(3,4) are two fixed points; P(x,y) divides AB internally in the ratio 1: m. Find the coordinates of P. If m is a variable, deduce from the above, equation of the straight line joining A and B.            {(m+3)/(m+1), (2m+4)/(m+1)}; x - y+1= 0

**34) A point moves in such a way that the sum of its distances from two fixed points (ae,0) and (-ae,0) is 2a, prove that the equation of its locus is x²/a² + y²/b² = 1 where b²= a²(1- e²).
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