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