SARASWATI
STRAIGHT LINES
XI - BATCH
1. Find the equation of the parallel to y-axis and at a distance:
(i) 3 units to the right of it
(ii)
5 units to the left of it.
2. Find the equation of the line parallel to x-axis and at a distance:
(i) 2 units above it
(ii)
3 units below the x-axis.
3. (a) Find the equation of the line parallel to the x-axis and passing through the point (0,2).
(b) Find the equation of the line which is parallel to y-axis and passes through the point (3,4).
4. Write the equation for the axes.
5. Find the equation of the line perpendicular to x-axis and passes through
(i) The origin
(ii)
the point (-1-1).
6. Find the equations for the line which passes through the point (-2-5) and is perpendicular to
y-axis.
7. Find the equation of a line through the origin which makes an angle with positive x-axis equal
to
(i) 45°
(ii)
135°.
8. Find the equations of the bisectors of the angles between the co-ordinate axes.
9. Determine the equation of a line
(i) Passing through the point (-2, 3) and having slope -4.
(ii) Passing through the point (-4, 3) with slope ½.
(iii) Passing through (2,2√3 and inclined with the x-axix at an angle of 75°
(iv) Passing through (0.0) with slope m.
(v) Passing through (-2, 3) and making an angle of 60° with the positive direction of x-axis.
10. Find the equation of a line
(i) Intersecting x-axis at a distance 3 units to the left of origin with slope -2.
(ii) Intersecting the y-axis at a distance of 2 units above the origin and making an angle of
30° with positive direction of axis.
11. Find the equation of the line passing through the point (2,3) and parallel to the join of (4,-5)
and
(-7, 3).
12. Find the equation of the line passing through (-3, 5) and perpendicular to the line through
the points (2, 5) and (-3, 6).
13. Find the equation of the right bisector of the segment joining the points (3, 4) and (-1, 2).
14. The vertices of a triangle ABC are A (-1,1), B (3, 5) and C (4, 3). Find the equation of
(i) the altitude through A and
(ii) the right bisector of- side BC.
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15. Find the equation of the line parallel to the line 3x – 4y +2 = 0 and passing through the point
(-2, 3).
16. Find the equation of the line parallel to 3x +4y+ 5= 0 and passing through the mid-point of
the line joining (6, -7) and (2,- 3).
17. Find the equation of the line perpendicular to the line x – 2y + 3 = 0 and passing through the
point (1, -2).
18. Find the equation of the line which is perpendicular to 5x -2y =7 and passes through the
mid-point of the line joining (2, 7) and (-4, 1).
19. (i) Find the equation of the line that has y- intercept - 3 and is perpendicular to the line 3x+
5y = 4.
(ii) Find the equation of a straight line perpendicular to the line x- 7y + 5 = 0 and having x –
intercept 3.
20. Find the equation of the line passing through the points:
(i)
(-1, 1) and (2, -4)
(ii) (0, -3) and (5, 0)
(iii)
(1,-1) and (3, 5)
(iv) (2, 3) and (5,-2)
21. Find the equation of the sides of the triangle whose vertices are (2, 1), (-2, 3) and (4, 5).
22. The vertices of ∆APQR are P(2, 1), Q(-2, 3) and R(4, 5). Find the equation of the median
through the vertex R.
23. Find the equation of the medians of the triangle whose vertices are (-1, 6), (-3,-9), (5,-8).
24. (a) Show that the point (1, 4), (3,-2) and (-3, 16) are collinear. Find equation of the line
through them.
(b) By using the concept of equation of a line, prove that the three points (3, 0), (-2,-2) and
(8, 2) are collinear.
25. The vertices of a quadrilateral are A(-2,6), B (1, 2), C (10, 4) and D (7, 8). Find the equation of
its diagonals.
26. (a) In what ratio, the line joining (-1, 1) and (5, 7) is divides by the line x+ y= 4 ?
(b) Find the ratio in which the line joining (2, 3) and (4, 1) divides the line joining (1, 2) and (4,
3).
27. Find the equation of a line equidistant from the lines y= 10 and y= -2.
28. The perpendicular from the origin to a line meets it at the point (-2, 9), find the equation of
the line.
29. Find the values of K for which the line (k – 3)x – (4 – k2)y + k2 -7k + 6 = 0 is
(i) parallel to x-axis
(ii) parallel to y- axis
(iii) passes through origin.
30.(i) Write down the equation of a straight line with slope 5 and y-intercept 5.
(ii) Determine the equation of a line which cuts off an intercept – 5 on y-axis and having
slope ½.
(iii) Find the equation of the line parallel to the x- axis and has intercept on the y-axis as -2.
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31. (i) Find the equation of the line which cuts off an intercept of 5 units from the negative
direction of the axis of y and makes an angle of 60° with the positive direction of the axis of
x.
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(ii) Write the equation of the line for which tan𝜃 = 2 , where 𝜃 is the inclination of the line
3
and (a) y-intercept − 2 (b) x-intercept 4.
(iii) Find the equation of the line which makes an angle cot-1 1/3 with y-axis and cuts off an
intercept of 5 units from – ve direction of y- axis.
32. Find the value of m and c in order that the point (3, 7) and (-2, 6) may satisfy the equation y=
mx +c.
33. (a) Find the equation of the line which makes intercepts – 3 and 2 on the x-axis and y-axis
respectively.
(b) Show that the straight line whose intercepts on the axes of x and y are 3 and -2
respectively passes through the point (9, 4).
34. Find the equation of the line which passes through the point (2, 3) and makes equal
intercepts on the axes of x and y.
35. Find the equation of the line which passes through the point (6, 7) and cuts off intercepts on
the axes which are equal in magnitude but opposite in sign.
36. (i) Find the equations of the lines passing through the point (6, -4) such that the sum of the
intercepts on the axes is 7.
(ii) Find the equation of the lines passing through the point (2, 2) such that the sum of its
intercepts on the axes is 9.
37. Find the equation of the line which passes through the point (1, -3) and makes an intercept
on the y-axis twice as long as that on the x-axis.
38. Find the equation of the line which passes through the point (3, -2) and cuts off positive
intercepts which are in the ratio 4 : 3.
39. (i) Find the equation of the line such that the segment of the line intercepted between the
axes is bisected by the point (1, -3).
(ii) P(a, b) is the mid-point of a line segment between the axes. Show that equation of the
line is x/a +y/b=2.
40. Point R(h, k) divides a line-segment between the axes in the ratio 1:2 Find equation of the
line.
41. Find the equation of the line which passes through the point (-4, 3) and is such that the
portion of it between the axes is divided by the point in the ratio 5 : 3.
42. Find the equation of the line which has length of perpendicular segment from the origin to
the line as 5 and the inclination of the perpendicular segment with the positive direction of
x-axis as 30°.
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43. Find the equation of a line where the perpendicular distance of the origin from the line is p
and the angle made by the perpendicular with the x-axis is ω :
(i) P = 3, ω = 45°
(ii) p = 5, ω = 30°
(iii) p = 5, ω= 135°
(iv) p = 1, ω = 90°
(v) p = 3, ω = 180°
44. Find the equation of the straight line at a distance 6 units from the origin such that the
perpendicular from the origin to the line makes an angle cot1 (12/2) with the positive
direction of x-axis.
45.Find the values of 𝜃and p, if the equation x cos𝜃+y sin𝜃= p is the normal form of the line
√3 𝑥 + 𝑦 + 2 = 0.
46. Reduce each of the following equation into slope intercept form and find their slope and the
y-intercepts.
(i) 3x+ 3y= 5
(ii) 7x+ 3y- 6 = 0
(iii) 2x- 4y= 5
(iv) 6x+ 3y -5 = 0
(v) x+ 7y= 0
(vi) y= 0.
47. Reduce the following equation to the intercept form. Also find the intercepts on the axes:
(i) 3x+ 2y- 12= 0
(ii) 4x- 3y = 6
(iii) 3y+ 2 = 0
48. Equation of a line is 3x- 4y+ 10. Find its (i) slope (ii) x- and y-intercepts.
49. Find the intercepts of the line x sin𝛼 + 𝑦𝑐𝑜𝑠𝛼 = sin 2𝛼 on the co-ordinate axes and obtain
the mid-point of the line segment intercepted between the axes.
50. Find the inclinations of the following lines with x-axis:
(i) X-√3y + 2 + 0.
(ii) 2x - 2y + 7= 0.
(iii) 2√3x + 6y+ 5 = 0.
51. Reduce each of the following to the normal form. Find the length of the perpendicular from
the origin to the line and angle between the perpendicular and the positive x-axis.
(𝑖𝑖) 𝑥 − √3𝑦 + 8 = 0
(i) √3𝑥 + 𝑦 − 8 = 0
(iii) 4x + 3y – 9 = 0
(iv) x – y = 4
(V) x – 4 = 0
vi) y – 2 = 0.
52. 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.
53. Find the equations of the straight lines passing through the point (3, -2) and inclined at 60° to
the straight line √3𝑥 + 𝑦 = 1.
54. Find the equation of the line parallel to the line 3x - 4y + 2= 0 and passing through the point
(-2, 3).
55. Find the equation of the line parallel to 3x + 4y + 5 = 0 and passing through the mid-piont of
the line joining (6, -7) and (2, -3).
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56. Find the equation of the line perpendicular to the line x – 2y + 3 =0 and passing through the
point (1, -2).
57. Find the equation of the line which is perpendicular to 5x – 2y = 7 and passes through the
mid-point of the line joining (2, 7) and (-4, 1).
58. (i) Find the equation of the line that has y- intercept – 3 and is perpendicular to the line 3x +
5y = 4.
(ii) Find the equation of a straight line perpendicular to the line x -7y + 5 = 0 and having xintercept 3.
59. Find the point of intersection of the lines.
(i) 2x + 3y – 6 = 0, 3x – 2y – 6 = 0 (ii) x = 0, 2x – y + 3 = 0
(ii) x/3 – y/4= 0, x/2 + y/3 = 1.
60. (i) The sides of a triangle are given by x - 2y + 9 = 0,3x + y – 22 = 0 and x + 5y + 2 = 0. Find the
vertices of the triangle.
(ii) Find the equation of the line parallel to y-axis and drawn through the point of intersection
of the lines x – 7y + 5 = 0 and 3x + y = 0.
61. Prove that the following lines are concurrent:
(i) 2x + 3y - 13 = 0, x + 2y- 8 = 0 and 3 x – y – 3 = 0
(ii) 4x+ 3y= 8, x + y = 1 and 4 x + 5 y = 0
(iii)x/a + y/b = 1, x/b + y/a = 1 and x - y = 0.
62.Prove that the following lines are concurrent and find the point of concurrence in each case:
(i) 5x – 3y = 1, 2x + 3y = 23, 42x + 21y = 257
(ii) 2x + 3y – 4 =0, x - 5y + 7 = 0, 6x – 17y + 24 = 0.
63. Find the value of k so that the three lines 2x + y – 3 = 0, 5x + ky – 3 = 0 and 3x – y – 2 = 0 are
concurrent.
64. Find the value of p so that the three lines 3x + y – 2 = 0, px + 2y – 3 = 0 and 2x – y – 3 = 0 may
intersect at one point.
65. If the lines whose equation are = m1x + c1, y = m2x + c2, and y= m3x + c3 meet in a point, then
prove that m1 (c2-c3) + m2(c3-c1) +m3(c1-c2) = 0.
66. Two lines cut the axis of x at distances 4 and -4 and the axis of y at distances 2 and 6
respectively. Find the co-ordinates of their point of intersection.
67. (a) Find the area of the triangle formed by the lines:
(i) y –x = 0, x + y = 0 and x - k = 0
(ii) x – y +1 = 0, 7x – 4y + 1 = 0 and 8x – 5y + 1 = 0
(iii) x + 4y =9, 9x + 10y + 23 = 0 and 7x + 2y = 1.
(b) Show that the area of the triangle formed by the lines y= m1x + c1, y = m2x + c2 and
x= 0 is (c1-c2)2
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2IM1 – M2I
68. Show that the straight lines 7x – 2y + 10 = 0, 7x + 2y – 10 = 0 and y+2 = 0, form an isosceles
triangle whose area is 14 sq. units.
69. Find the centroid of the triangle formed by the lines 3x – y – 11 = 0, 7y + x – 11 = 0 and 2x +
3y = 0.
70. The co-ordinates of the points A,B and C are (1, 2), (-2, 1) and (0, 6). Verify that the medians
of trangle ABC are concurrent. Also find the co-ordinates of the point of concurrence
(centroid).
71. (i). Find the co-ordinates of the foot of the perpendicular from the point (3, -3) on the line x2y = 4.
(ii) Find the co-ordinates of the foot of the perpendicular from the point(-1, 3) to the line 3x 4y – 16 = 0.
72.The three sides of AB,BC and CA of a triangle ABC are 5x – 3y + 2 = 0, x – 3y -2 = 0 and x + y –
6 = 0 respectively. Find the equation of the altitude through vertex A.
73. Find the co-ordinates of the foot the perpendicular from the point (-3, -5) on the line x + y - 6
= 0. Hence, find the distance of the point from this line.
74.(i)Find the image of the point (2, 3) w.r.t. the line mirror 2x + y + 3 = 0.
(ii) Find the image of the point (1, 2) w.r.t. line mirror x-axis.
75. Find the co-ordinates of the orthocenter of the triangle whose vertices are (-1, 3), (2, -1) and
(0, 0).
76. A triangle is formed by the line y + x – 6 = 0, 3y – x + 2 = 0 and 3y = 5x + 2. Find the coordinates of its orthocenter.
77. Two vertices of a triangle ABC are A(5, -1) and B (-2, 3). If the orthocenter of the triangle is
the origin ; find the co-ordinates of the third vertex.
78.Assuming that straight lines work as the plane mirror for a point, find the image of the point
(1, 2) in the line x – 3y + 4 = 0.
79. If the image of the point (2, 1) w.r.t. a line mirror be (5, 2) ; find the equation of the mirror.
80.(i) Find the equation of a line drawn perpendicular to the line x/4 + y/6 = 1 through the point
where it meets the y-axis.
(ii) Find the equation of a straight line drawn perpendicular to the line x/a + y/b = 1 through
the point where it meets the y-axis.
81. Find the distance of the line 4x – y = 0 from the point P(4, 1) measured along the line making
an angle of 135° with the positive x- axis.
82.Find the distance of the line 4x – y = 0 from the point p(4, 1) measured along the lines 4x + 7y
– 3 = 0 and 2x - 3y + 1 = 0, that has equal intercepts on the axes.
83. Find the distance of the line 4x + 7y + 5 = 0 from the point (1, 2) along the line 2x – y = 0.
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84. Find the direction in which a straight line must be drawn through the point (-1, 2) so that its
point of intersection with the line x + y = 4 may be at a distance of 3 units from this point.
85. Find the length of the perpendicular drawn from
(i) the point (3, -5) on the line 3x – 4y – 26 = 0.
(ii) the point (-3, 4) on the line 3x + 4y – 5 = 0.
(iii)the point (3, -1) on the line 12x – 5y – 7= 0.
86. Find the distance between the line 3x – 4y + 12 = 0 and the point (4, 1).
87. In the triangle with vertices. A (2, 3), B (4, -1) and C (1, 2); find the equation and length of the
altitudes from vertex A.
88. The vertices of a triangle are A(-2, 1), B(6, -2) and C(4, 3). Find the lengths of the altitudes of
the triangle.
89. Find the distance between the parallel lines
(i) 3x – 4y + 7 = 0 and 3x – 4y + 5 = 0
(ii) 4x – 3y – 9 = 0 and 4x – 3y – 24 =0
(iii) 3x + 4y – 5 = 0 and 6x + 8y – 45 = 0
(iv) 15x + 8y – 34 = 0 and 15x + 8y + 31 = 0
(v) y = mx + c and y = mx + d.
(vi) l(x + y) + p = 0 and lx + ly – r = 0.
90. Find the length of perpendicular from (-2, 3) to the line y=2x – 3 and also find the coordinates of the foot of the perpendicular.
91. 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.
92. Prove that the product of the lengths of the perpendicular drawn from the points (√𝑎2 − 𝑏 2,
𝑥
𝑦
0) and(-√𝑎2 − 𝑏 2 , 0) to the line 𝑎 cos𝜃 + 𝑏 𝑠𝑖𝑛𝜃 = 1 is b2.
93. Find the lines parallel to 3x + 4y – 10 = 0 and at a distance 5 on either side of it.
𝑥
𝑦
94. What are the points on the y-axis whose distance from the line 3 + 4 =1 is 4 units?
95. Find the points on the line y = x which are at a distance of 5 units from 4x + 3y – 1 = 0.
96.Find the equation of two straight lines parallel to 3x + 4y + 1 =0 and at a distance of 2 units
from the point (-1, 2).
97. Find the equation of the two line which are parallel to the line through (1, 2) and (5, 5) and at
a distance of 3 units from it.
98. Find the equations of the lines drawn through the point (0, 1) on which the perpendiculars
dropped from the point (2, 2) are each of unit length.
99. Show that the points (-3, -2) and (-1, 1) lie on the same side of the line 7x - y + 2 = 0.
100. Show that the points (-2, 5) and (3, -1) lie on the opposite sides of the line 5x - 3y + 7 = 0.
101. If sum of the perpendicular distances of a variable point p(x, y) from the lines x + y – 5 = 0
and 3x – 2y + 7 = 0 is always 10, show that P must move on a line.
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102. Find equation of the line which is equidistant from parallel lines 9x + 6y – 7 = 0 and 3x + 2y +
6 = 0.
103.(a) Find the equation of the straight line which divides internally the line joining A (-3, 7)
and B (5, -4) in the ratio 4: 7 and which is perpendicular to AB.
(b) A line perpendicular to the line segment joining the points (1, 0) and (2, 3) divides it in
the ratio 1: n. Find the equation of the line.
104.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
105.Find the equation of the two straight lines passing through the point (3, -2) and inclined
at 60° to the straight line √3𝑥 + 𝑦 = 1.
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