Class XI Maths
Ch. 10. Straight Lines
Question Bank Part III
1. Find the equation of the line passing through:
a) (– 2, 5) and (8, 7)
b) ( 3, – 1) and (– 4, – 5)
2. Find the equation of the line:
a) passing through (3, 2) and having slope – 1/3
b) making intercepts – 2/3 and – 4/3 on the axes.
c) passing through (– 1, 6) and making an angle of 150⁰ with the positive x – axis.
3. Find the value of p such that the line passing through (– 4, p) and (1, 3) is : a) parallel b) perpendicular
to the line passing through the points (– 2, 5) and (8, 7).
4. For what values of x, the area of the triangle formed by the points (5, – 1), (x, 4) and (6, 3) is 5.5 sq. units?
5. Show that the points (– 1, 2), (5, 0) and (2, 1) are collinear by using
a) distance formula b) area formula c) slope formula.
6. Find the value of m and c so that the line with the equation y = mx + c may pass through the points
(– 2, 3) and (4, – 3).
7. Find the equation of the line passing through (– 4, – 5) and perpendicular to the line passing through the
points (– 2, 3) and (4, – 3).
8. The mid points of the sides of a triangle are (2, 2), (2, 3) and (4, 6). Find the vertices and the equation of
the sides of the triangle.
9. Find the equation of the perpendicular bisector of the line segment joining the points (0, 3) and (– 4, 1).
10. Find the angle between the lines joining the points (3, – 1) to (2, 3) and (2, 7) to (5, 12).
11. Find the equation in normal form:
a) p = 3 ; ω = 315⁰
b) p = √3 ; ω = 240⁰ c) p = 1 ; ω = – 60⁰
12. Find the co-ordinates of the circumcentre of the triangle, whose vertices are (8, 6), (8, -2) and (2, -2). Also
find its circumradius.
13. Find the coordinates of the orthocenter of the triangle formed by the lines x + y – 6 = 0 and 3y = 5x + 2.
14. Write the equation of a line which has the y – intercept 2 and slope 7.
15. Write the equation of a line which has the y – intercept – 1 and is parallel to y = 5x – 7 .
16. Write the equation of a line which has the y – intercept – 5 and is equally inclined to the axis.
17. What will be the value of m and c, if the straight line y = mx + c passes through the points (3, -4) and
(-1, 2)?
18. Find the equation of a line, which has y – intercept 2 and is perpendicular to the y = 2x + 1.
19. Find the equation of a line whose y – intercept is – 4 and which is parallel to the line joining the points
(-3, 4) and ( 2, -5).
20. Find the equation of a line whose y – intercept is 5 and which is perpendicular to the line joining the points
(1, -2) and (-2, -3).
21. Find the distance of the point A(2, 3) from the line 2x - 3y + 9 = 0 measured along a line making an angle
of 450 with X axis.
22. Find the angle between the lines y  (2  3 ) x  9 and y  (2  3 ) x  1 .
23. Find the equation of the straight line passing through (4, - 2) and making an angle of 60⁰ with the negative
direction of Y axis.
24. Reduce x – y + 22 into normal form and hence find the value of  and P.
25. Find the image of the point (1, - 2) on the line y = 2x + 1.
26. A ray of light is sent along the line x - 2y - 3 = 0. Upon reaching the line 3x - 2y - 5 = 0 the ray is reflected
from it. Find the equation of the line containing the reflected ray.
27. Find the equation of the line mid parallel to the lines 9x + 12y - 15 = 0 and 3x + 4y - 15 = 0
28. One side of a rectangle lie along the line 4x + 7y +5 = 0. Two of its vertices are (-3,1) and (1,1). Find the
equation of the diagonals of the rectangle.
29. Find the equation of the bisector of the angle A of the triangle ABC whose vertices are A(-2, 4), B(5,5)
and C(4, -2).
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30. Show that the line joining (-2, 6) and (4, 8) is perpendicular to the line joining (8, 12) and (4, 24).
31. Find for what values of x the area of the triangle formed by the points (5, - 1), (x, 4) and (6, 3) is 5.5
square units?
32. Find the equation of a line passing through the intersection of the lines x - 3y + 1 = 0 and 2x + 5y – 9 = 0
and whose distance from the origin is 2 units.
33. Find the ratio in which the line joining the points (2, 3) and (4, 1) divides the line joining (1, 2) and (4,3).
Also find the point of intersection.
34. Find the equation of a line perpendicular to 5x - 2y = 7 and passing through the midpoint of the line
joining (4, -1) and (2, 5).
35. Determine the equation of a line passing through (4, 5) and making equal angles with the lines
5x - 12y + 6 = 0 and 3x = 4y + 7.
36. A line cuts x axis at A and y axis at B. The point (2, 2) divides AB in the ratio 2:1.Find the equation of the
line.
37. Find the equation of a line passing through the point of intersection of the lines 5x - 3y = 1 and
2x + 3y = 23 and perpendicular to the line x - 2y = 3.
38. Find the equation of a line passing through (3, - 2) and inclined at an angle 600 to the line 3 x + y =1
1

39. If the angle between two lines is and slope of one line is , find the slope of the other line.
2
4
40. The slope of one line is double the slope of the other. If the tangent of the angle between them is 1/3,find
their slopes.
a b
41. If three points (h, 0), (a, b) and (0, k) lie on a line show that   1 .
h k
42. A line perpendicular to the line joining the points (2, - 3) and (1, 2) divides it in the ratio 1:2. Find the
equation of the line.
43. Find the equation of a line passing through (2, 2) and cutting of intercepts whose sum is 9
44. The vertices of a triangle PQR are P(-2, 3), Q(2, 5) and R(-1 ,3) . Find the equation of the median and
altitude through R.
45. Express x - 3 y + 8 = 0 in the normal form.
46. If a and b are the lengths of the perpendiculars from the origin to the lines x cosθ - y sinθ = k cos2θ and
x secθ + y cosecθ = k ,respectively, prove that a2 + 4b2 = k2.
47. If p is the length of the perpendicular from the origin to the line whose intercepts on the axes a and b, then
1
1
1
show that 2  2  2 .
p
a
b
48. Find the angle between the lines
3x  y  1and x  3 y  1.
49. Two lines passing through the points (2, 3) intersect each other at an angle of 600. If slope of one line is 2,
find the equation of the other line.
50. In triangle ABC with vertices A(2, 3) , B(4, 2)and C(1, 2), find the equation and length of the altitude
from the vertex A.
51. The line through the points (a, 3) and (4, 1) intersects the line 7x - 9y - 19 = 0 at right angles. Find the
value of a.
52. If the lines 2x + y - 3 = 0 , 5x + ky - 3 = 0 and 3x – y – 2 = 0 are concurrent, find value of k.
(c1  c 2 ) 2
53. Show that the area of the triangle formed by the lines y = m1x + c1 , y = m2x + c2 and x = 0 is
.
2 m1  m2
54. Find the equation of the lines through the points (3, 2) which make an angle of 45o with the line x - 2y = 3
55. 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.
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56. Show that the equation of the line passing through the origin and making an angle θ with the line
y m  tan 
y = mx + c is 
x 1  m tan 
57. In what ratio the line joining (-1, 1) and (5, 7) is divided by the line x + y = 4?
58. Find the distance of the line 4x + 7y + 5 = 0 from the point (1, 2) along the line 2x – y = 0.
59. Find the image of the point (3, 8) with respect to the line x + 3y = 7
60. Find equation of the line equidistant from the the parallel lines 2x + 3y + 5 = 0 and 4x + 6y + 7 = 0.
61. Prove that the product of the lengths of the perpendicular from the points
x
y
( a 2  b 2 ,0) and ( a 2  b 2 ,0) to the line cos   sin   1is b 2
a
b
62. A straight line passes through the point (2, 3) and its segment intercepted between the axes is bisected at
that point. Find its equation.
63. Find the distance of the point A(2, 3) from the line 3y = 2x + 9 measured along a line making angle 450
with the x axis.
64. Find the equation of a line such that the segment intercepted by the axes is divided by the point (-5, 4)
in the ratio 1:2.
65. Find equation of lines which cut off intercepts whose sum and product are 1 and -6 respectively.
66. Find the equation of a line passing through (-1, 2) and whose distance from the origin is 1 unit.
67. Find the equation of a line which is perpendicular to the line 4x – y + 8 = 0 and passes through the
midpoint of the line segment joining (1, 5) and (3, 11).
68. Find perpendicular distance from the origin of the line joining the points (cos θ, sin θ) and (cosφ, sinφ).
69. Find the value of K such that the line joining the points (2, K) and (-1, 3) is parallel to the line joining
(0, 1) and (-3, 1).
70. Show that points (a, b + c), (b, c + a), (c, a + b) are collinear.
71. Show that points (at12 , 2at1) , (at22 , 2at2) and (a , 0 ) are collinear if t1t2 = - 1
72. Find the equation of the line that has y intercept 4 and is parallel to the line 2x – 3y =7
73. Find the equation of the line that has x intercept - 3 and is perpendicular to the line 3x + 5y = 4.
74. Prove that the lines 7x – 2y + 5 = 0 and 14x – 4y – 8 = 0 are parallel to each other.
75. Prove that the lines 3x – 2y + 5 = 0 and 4x + 6y – 23 = 0 are perpendicular.
76. Find out the angle between the following pair of lines
a. y – √3x – 5 = 0 and √3y – x + 6 = 0
b. y = ( 2 – √3)x + 5 and y = ( 2 + √3)x – 2
77. Find the equation of a line which passes through the point (3, - 2) and is inclined at 600 to the line
√3x + y = 1.
78. Find the equation of a line which passes through the point (x1 , y1) and perpendicular to the line
xy1 + x1y = a2
79. A line such that its segment between the axis is bisected at the point (x1, y1). prove that the equation of the
line is
x
y

1
2 x1 2 y 2
80. If the three lines a1x + b1y = 1, a2x + b2y = 1 and a3x + b3y = 1 are concurrent, prove that the points
(a1,b1), ( a2,b2) and ( a3,b3) are collinear.
81. Find the value of k such that the three lines x + y – 3 = 0, kx – y – 5 = 0 and 3x + y – 7 = 0 are concurrent.
82. Find the length of the perpendicular drawn from the point (b, a) to the line
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x y
 1
a b
----------------------------------------------------------------------------------------------------------------------------------ANSWERS
1.
2.
3.
4.
5.
6.
7.
8.
a) x – 5y + 27 = 0
b) 4x – 7y – 19 = 0
a) x + 3y – 9= 0
b) 6x + 3y + 4 = 0
c) x + √3y – 6√3 + 1 = 0
a) p = 2
b) p = 28
x = 9 or 7/2
Find AB, BC and AC…..sum of any two distances should be equal to third distance.
Substitute the coordinates for x and y to form two equations . solve to get m = – 1 and c = 1.
x–y–1=0
vertices are (4, 5), (4, 7) and (0, - 1).
Equation of the sides are x = 4, 3x – 2y – 2 = 0 and 2x – y – 1 = 0
9. perpendicular bisector passes thru the mid- point, then use m1.m2 = – 1 ans: 2x + y + 2 = 0.
10. 45⁰
11. a) x – y = 3√2
b) x + √3y + 2√3 = 0
c) x – √3y – 2 = 0
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