EXERCISE--1
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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
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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
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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)
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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)
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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
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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
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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
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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
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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
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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
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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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