DGT Straight Line
DGT Straight Line
DGT Straight Line
Straight Line 57
6 Straight Line
y intercept
2. Equation of straight line in standard forms: c
i. Slope intercept form: The equation of a b
line with slope ‘m’ and making an intercept iii. Normal form : To change the general form
on Y – axis is y = mx + c of a line into normal form, first take c to
ii. Slope - point form: The equation of a line right hand side and make it positive, then
which passes through the point (x1, y1) and divide the whole equation
has slope ‘m’ is y – y1 = m (x – x1)
iii. Two point form: The equation of a line
by a 2 b 2 like
passing through two points (x1, y1) and ax by c
,
y y1 x x1 a 2 b2 a 2 b2 a 2 b2
(x2,y2) is y y x x
1 2 1 2
a
where cos ,
iv. Double - intercept form: The equation of a b2
2
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Straight Line 58
4. Two intersecting lines:
ax1 by1 c
i. Angle between two intersecting lines: is given by p
a. If is the acute angle between the lines a 2 b2
with slopes mi and m2, then Length of perpendicular from origin to the
m1 m 2 c
tan line ax + by + c = 0 is
1 m1m 2 a b2
2
b. If is the acute angle between the lines ii. Distance between two parallel lines :
a1x + b1y + c1 = 0 and + c2 = 0, and The distance between two parallel lines
a2x + b2y + c2 = 0, then a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0
a 2 b1 a1b 2 c1 c 2
tan is
a1a 2 b1b 2 a 2 b2
ii. Point of intersection of two lines: Shortcuts
Point of intersection of two lines
1. Slope of the line equally inclined with the axis is
a1x + b1y + c1 = 0 and a2x + b2 y + c2 = 0 is 1 or – 1
b c b 2 c1 c1a 2 c 2 a1 2. Slope of two parallel lines are equal.
(x ', y') 1 2 , 3. Equation of a line parallel to ax + by + c = 0 is
a1b 2 a 2 b1 a1b 2 a 2 b1
ax + by + = 0, where , is a constant.
5. Concurrent lines: 4. Equation of a line perpendicular to ax + by + c = 0
Three or more lines are said to be concurrent if is bx – ay + , = 0, where , is a constant.
they meet at a point. 5. If the equation of line be a sin + b cos = c,
i. If the three lines a 1x + b 1y + c 1 = 0, then line
a2x + b2y + c2 = 0 and a3x + b3y + c3 = 0 are i. Parallel to it is a sin + b cos = d
concurrent, then a 1 (b 2 c 3 – b 3 c 2) – b 1
ii. Perpendicular to it is a
(a2c3 – a3c2) + c1 (a2b3 – a3b2) = 0
6. Length of perpendicular:
sin b cos d
i. Distance of a point from a line : 2 2
The length p of the perpendicular from the
point (x , y ) to the line ax + by + c = 0
1 1
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Straight Line 59
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Straight Line 60
16. The equation of the line which cuts off intercepts 24. The lines a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0
2a sec and 2a cosec on X-axis and Y-axis are perpendicular to each other, if
respectively is a) a1b2 – b1a2 = 0
a) x sin + y cos – 2a = 0 b) a1a2 + b1b2 = 0
b) x cos + y sin – 2a = 0
c) a12 b2 b12 a 2 0
c) x sec + y cosec – 2a = 0
d) a1b1 + a2b2 = 0
d) x cosec + y sec – 2a = 0
17. A straight line makes an angle of 135° with the 25. The line passing through (1, 0) and ( 2, 3)
X-axis and cuts Y-axis at a distance – 5 from the makes an angle of _____ with X-axis.
origin. The equation of the line is a) 600 b) 1200
a) 2x + y + 5 = 0 b) x + 2y + 3 = 0 c) 1500 d) 1350
c) x + y + 5 = 0 d) x + y + 3 = 0
6.2 Two intersecting lines and family of lines
18. The equation of line perpendicular to x + c is
a) y = d b) x = d 26. The equation of a line through the intersection of
lines x – 0 and y = 0 and through the point (2, 2)
c) x = 0 d) None of these
is
x y a) y = x – 1 b) y = – x
19. If the line 1 passes through the points a
a b c) y = x d) y = – x + 2
(2, – 3) and (4, – 5), then (a, b) = 27. For the lines 2x + 5y = 1 and 2x – 5y = 9, which
a) (1, 1) b) (– 1, 1) of the following statement is true?
c) (1, –1) d) (– 1, – 1) a) Lines are parallel
20. The equation of a line perpendicular to line b) Lines are coincident
ax + by + c = 0 and passing through (a, b) is c) Lines are intersecting
equal to d) Lines are perpendicular
a) bx – ay = 0 b) bx + ay – 2ab = 0 28. The acute angle between the lines y = 3 and
c) bx + ay = 0 d) bx – ay + 2ab = 0
y 3x 9 is
21. A line passes through (2, 2) and is perpendicular
to the line 3x + y = 3. Its y – intercept is a) 300 b) 600
c) 450 d) 900
1 2
a) b) 29. The angle between the lines y (2 3)x 5
3 3
4 and y (2 3)x 7 is
c) 1 d) a) 300 b) 600
3
22. The number of straight lines which are equally c) 450 d) 900
inclined to both the axes is 30. Equations of the two straight lines passing through
a) 4 b) 2 the point (3, 2) and making an angle of 450 with
the line x – 2y = 3, are
c) 3 d) 1
a) 3x + y + 1 = 0 and x + 3y + 9 = 0
23. Let P(–l, 0), Q(0, 0) and R(3, 3 3 ) be three b) 3x – y – 7 = 0 and x + 3y – 9 = 0
points. Then, the equation of the bisector of the c) x + 3y – 1 = 0 and x + 3y – 9 = 0
angle PQR is
d) None of these
3 31. The value of k for which the lines 7x – 8x + 5= 0,
a) xy0 b) x 3y 0 3x – 4y + 5 = 0 and 4x + 5y + k = 0 are concurrent
2
is given by
3 a) – 45 b) 44
c) 3x y 0 d) x y0
2 c) 54 d) – 54
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Straight Line 61
32. The lines 15x – 18y + 1 = 0, 12x + 10y – 3 = 0 39. The length of the perpendicular from the point
and 6x + 66y – 11 = 0 are x y
(b, a) to the line 1 is
a) Parallel b) Perpendicular a b
c) Concurrent d) None of these
a 2 ab b 2 b 2 ab a 2
33. If u = a1x + b1y + c1 = 0, v = a2x + b2y + c2 = 0 a) b)
a 2 b2 a 2 b2
a b c
and 1 1 1 , then the curve u + kv = 0 is
a 2 b 2 c2 a 2 ab b 2 a 2 ab b 2
a) same straight line u c) d)
a 2 b2 a 2 b2
b) different straight line
40. The length of perpendicular drawn from origin
c) not a straight line on the line joining (x', y') and (x'', y'') is
d) None of these
34. The angle between the lines whose intercepts on x ' y '' x '' y'
a)
the axes are (a, – b) and (b, – a) respectively, is (x '' x ') 2 (y '' y ')2
a 2 b2 b2 a 2
a) tan 1 b) tan 1 x ' y'' x ''y '
ab 2 b)
(x '' x ') 2 (y '' y ')2
b2 a 2
c) tan 1 d) None of these
2ab x ' y '' y ' y ''
35. The angle between the lines c)
(x '' x ')2 (y '' y ') 2
x cos 300 + y sin300 = 3 and x cos 600 + y sin 600 = 5
is
x ' x '' y ' y ''
a) 900 b) 300 d)
(x '' x ')2 (y '' y') 2
c) 600 d) 450
36. If the lines y = m 1x + cl, y = m2x + c2 and 41. The perpendicular distance of the straight line
y = m3x + c3, are concurrent, then 12x + 5y = 7 from the origin is
a) m1(c2 – c3) + m2(c3 – c1) + m3(c1 – c2) = 0 7 12
a) b)
B) m1(c2 – c1) + m2(c3 – c2) + m3(c1 – c3) = 0 13 13
c) c1(m2 – m3) + c2(m3 – m1) + c3(m1 – m2) = 0 5 1
d) C1(m1 – m2) + c2(m2 – m3) + C3(m3 – m1) = 0 c) d)
13 13
37. The angle between the lines a1x + b1y + c1 = 0 42. The length of perpendicular from (3, 1) on line
and a2x + b2y + c2 = 0, is 4x + 3y + 20 = 0, is
1 a1b 2 a 2 b1 a1b 2 b1b 2 a) 6 b) 7
a) tan b) cot 1
a1a 2 b1b 2 a1b 2 a 2 b1 c) 5 d) 8
43. The distance of the point (– 2, 3) from the line
a1 b 2 a 2 b 2 a1b1 a 2 b2
c) cot 1 d) tan 1 x – y = 5 is
a1a 2 b1b 2 a1a 2 b1b2
a) 5 2 b) 2 5
6.3 Distance of a point from a line
c) 3 5 d) 5 3
38. If the length of the perpendicular drawn from the
origin to the line whose intercepts on the axes 44. Distance between the lines 5x + 3y – 7 = 0 and
are a and b be p, then 15x + 9y + 14 = 0 is
1 35 1
b) a b
2 2
a) a2 + b2 = p2 a) b)
p2 34 3 34
1 1 1 35 35
1 1 2 2 2 c) d)
c) d) 2 3 34 2 34
a 2 b2 p2 a b p
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Straight Line 62
45. The position of the point (8, – 9) with respect to 5. The equation of the line parallel to the line
the lines 2x + 3y – 4 = 6 and 6x + 9y + 8 = 0 is 2x – 3y = 1 and passing through the middle point
a) Point lies on the same side of the lines of the line segment joining the points (1, 3) and
(1, – 7), is
b) Point lies on the different sides of the line
a) 2x – 3y + 8 = 0
c) Point lies on one of the line
b) 2x – 3y = 8
d) None of these
c) 2x – 3y + 4 = 0
46. The length of perpendicular from the point
d) 2x – 3y = 4
(a cos , a sin ) upon the straight line
y = x tan + c, c > 0 is 6. The intercept cut off from Y-axis is twice that
from X-axis by the line and line passes through
a) c cos b) c sin2 (1, 2), then its equation is
c) c sec 2
d) c cos2 a) 2x + y = 4 b) 2x + y + 4 = 0
47. The equations (b – c)x + (c – a) y + (a – b) = 0 c) 2x – y = 4 d) 2x – y + 4 = 0
and (b3 – c3)x + (c3 – a3) y + a3 – b3 = 0 will
7. The equation of the straight line passing through
represent the same line, if the point (a cos3 , a sin3 ) and .perpendicular
a) b = c b) c = a to the line x sec + y cosec = a is
c) a = b d) All the above a) x cos – y sin 6 = a cos 2
b) x cos + y sin = a cos 2
Critical Thinking
c) x sin + y cos 0 = 3 cos 2
6.1 Slope of a line, Equation d) None of these
of a line in different forms 8. Equation of the line passing through the point
1. Equation of the hour hand at 4 O’ clock is (– 4, 3) and the portion of the line intercepted
between the axes which is divided internally in
a) x 3y 0 = b) 3x y 0 the ratio 5 : 3 by this point, is
a) 9x + 20y + 96 = 0
c) x 3y 0 d) 3x y 0
b) 20x + 9y + 96 = 0
2. A straight line through origin bisects the line c) 9x – 20y + 96 = 0
passing through the given points (a cos, a sin) d) None of these
and (a cos, a sin), then the lines are
9. A straight line through P (l, 2) is such that its
a) Perpendicular intercept between the axes is bisected at P. Its
b) Parallel equation is
a) x + 2y = 5
c) Angle between them is b) x – y + l = 0
4
c) x + y – 3 = 0
d) None of these
d) 2x + y – 4 = 0
3. A line L is perpendicular to the line 5x – y = 1 and
10. The point P(a, b) lies on the straight line
the area of the triangle formed by the line L and
3x + 2y = 13 and the point Q (b, a) lies on the
coordinate axes is 5. The equation of the line L is
straight line 4x – y = 5, then the equation of line
a) x + 5y = 5 b) x 5y 5 2 PQ is
a) x – y = 5 b) x + y = 5
c) x - 5y = 5 d) x 5y 5 2 c) x + y = – 5 d) x – y = – 5
4. A line passes through the point (3, 4) and cuts off 11. A line AB makes zero intercepts on X - axis and
intercepts from the co-ordinates axes such that Y - axis and it is perpendicular to another line
their sum is 14. The equation of the line is CD, 3x + 4y + 6 = 0. The equation of line AB is
a) 4x – 3y = 24 b) 4x + 3y = 24 a) y = 4 b) 4x – 3y + 8 = 0
c) 3x – 4y = 24 d) 3x + 4y = 24 c) 4x – 3y = 0 d) 4x – 3y + 6 = 0
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Straight Line 63
12. The line passing through (–1, /2) and
1
4 a) b) 2
3 sin 2cos is 2
perpendicular to
r
1
a) 2 3 r cos 2r sin c) d) – 2
2
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Straight Line 64
25. A line through A(– 5, – 4) meets the lines 31. The equation of straight line passing through the
x + 3y + 2 = 0, 2x + y + 4 = 0 and x – y – 5 = 0 at intersection of the lines x – 2y = 1 and x + 3y = 2
B, C and D respectively. If and parallel to 3x + 4y = 0 is
2 2 2
a) 3x + 4y + 5 = 0 b) 3x + 4y – 10 = 0
15 10 6
, then the equation c) 3x + 4y – 5 = 0 d) 3x + 4y + 6 = 0
AB AC AD
32. The point of intersection of the lines
of the line is
x y y y
a) 2x + 3y + 22 = 0 1 and 1 lies on the line
a b b a
b) 5x – 4y + 7 = 0
a) x – y = 0
x) 3x – 2y + 3 = 0
b) (x + y) (a + b) = 2ab
d) None of these
c) (lx + my) (a + b) = (l + m) ab
26. In what direction a line be drawn through the point
(1, 2) so that its points of intersection with the d) All of these
33. If the co-ordinates of the vertices A, B, C of the
6 triangle ABC are (– 4, 2), (12, – 2) and (8, 6)
line x + y = 4 is at a distance from the given
3 respectively, then B
point
1 6 6
a) 300 b) 450 a) tan b) tan 1
7 7
c) 600 d) 750
27. The sides AB, BC, CD and DA of a quadrilateral 1 7 1 7
are x + 2y = 3, x=1 , x – 3y = 4, 5x + y + 12 = 0 c) tan d) tan
6 6
respectively. The angle between diagonals AC
and BD is 34. If the lines y = 3x + 1 and 2y = x + 3 are equally
inclined to the line y = mx + 4, then m =
a) 450 b) 600
c) 900 d) 300 1 3 2 1 3 2
a) b)
6.2 Two intersecting lines and family of lines 7 7
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Straight Line 65
38. Which of the following lines is concurrent with 46. The equation of a line passing through the point
the lines 3x + 4y + 6 = 0 and 6x + 5y + 9 = 0 ? of intersection of lines x + 2y + 3 = 0 and
a) 2x + 3y + 5 = 0 3x + 4y + 7 = 0 and perpendicular to the line
x – y + 9 = 0 is
b) 3x + 3y + 5 = 0
a) x + y + 2 = 0 b) x – y – 2 = 0
c) 1x + 9y + 3 = 0
c) x + y – 5 = 0 d) x + 2y – 5 = 0
d) None of these
47. Three sides of a triangle are represented by the
39. The straight lines x + 2y – 9 = 0, 3x+ 5y – 5 = 0
equation x + y – 6 = 0, 2x + y – 4 = 0 and
and ax + by – 1 = 0 are concurrent, if the straight
x + 2y – 5 = 0. The co-ordinate of its orthocentre is
line 35x – 22y +1 = 0 passes through the point
a) (10, 11) b) (2, 3)
a) (a, b) b) (b, a)
c) (– 2, – 3) d) (– 11, – 10)
c) (– a, – b) d) (a, – b)
48. The equations of two equal sides of an isosceles
40. If the lines ax + y + 1 = 0, x + by + 1 = 0 and
triangle are 7x – y + 3 = 0 and x + y – 3 = 0 and
x + y + c = 0 (a, b, c being distinct and different
the third side passes through the point (1, – 10).
from 1) are concurrent, then
The equation of the third side is
1 1 1 a) 3x – y – 31 = 0 or x + y + 7 = 0
1 a 1 b 1 c b) 3x – y + 7 = 0 or x + 3y – 31 = 0
a) 0 b) 1 c) 3x + y + 7 = 0 or x – 3y – 31 = 0
1 d) Neither 3x + y + 7 nor x – 3y – 31 = 0
c) d) 3abc 49. For the straight lines given by the equation
abc
(2 + k) x + (1 + k)y = 5 + 7k, for different values
41. If the lines ax + 2y + 1 = 0, bx + 3y + 1 = 0 and
of k which of the following statements is true
cx + 4y + 1 = 0 are concurrent, then a, b, c are in
a) Lines are parallel
a) A. P. b) G. P.
b) Lines pass through the point (– 2, 9)
c) H.P. d) None of these
c) Lines pass through the point (2, – 9)
42. The straight lines 4ax + 3by + c = 0, where
a + b + c = 0, will be concurrent, if point is d) None of these
a) (4, 3) b) (1/4, 1/3) 50. The diagonals of the parallelogram whose sides are
lx + my + n = 0, lx + my + n' = 0, mx + y + n = 0,
c) (1/2, 1/3) d) None of these
mx + ly + n' = 0 include an angle
43. The value of , for which the lines 3x + 4y = 5,
5x + 4y = 4 and x + 4y = 6 meet at a point is
a) b)
a) 2 b) 1 3 2
c) 4 d) 3
1 l m
2 2
2lm
44. The equation of a line passing through the point c) tan 2 d) tan 1 2
l m 2 l m 2
of intersection of lines 2x + 3y + 1 = 0 and
3x – 5y – 5 = 0 and making an angle of 450 with 51. The opposite angular points of a square are (3, 4)
positive X-axis is and (1, –1). Then the co-ordinates of other two
a) 2x – 19y + 23 = 0 points are
b) 19x – 23y + 15 = 0
1 9 1 5
a) D , , B ,
c) 19x – 19y – 23 = 0 2 2 2 2
d) 20x – 19y + 23 = 0
45. Which of the following represents the equation 1 9 1 5
b) D , , B ,
of a line passing through point of intersection of 2 2 2 2
lines x + 2y + 5 = 0 and 3x + 4y + 1 = 0 and
9 1 1 5
passing through point (3, 2)? c) D , , B ,
2 2 2 2
a) 2x + 3y – 5 = 0 b) 3x + 2y – 13 = 0
c) x + 3y + 13 = 0 d) 3x – 2y – 7 = 0 d) None of these
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Straight Line 66
2 2
52. If a + b + c = 0 and p 0 the lines a) a b) b
2 2
ax + (b + c) y = p, bx + (c + a) y = p and c) a + b d) a2 – b2
cx + (a + b) y = p 60. The ratio in which the line 3x + 4y + 2 = 0 divides
the distance between 3x + 4y + 5 = 0 and
a) do not intersect b) intersect
3x + 4y – 5 = 0, is
c) are concurrent d) none of these
a) 5 : 3 b) 3 : 7
53. The equation of straight line passing through point
c) 2 : 3 d) None of these
of intersection of the straight lines 3x – y + 2 = 0
and 5x – 2y + 7 = 0 and having infinite slope is 61. If 2p is the length of perpendicular from the origin
a) x = 2 b) x + y = 3 x y
to the lines 1 , then a2, 8p2, b2 are in
c) x = 3 d) x = 4 a b
54. If vertices of a parallelogram are respectively a) A. P. b) G.P.
(0, 0), (1, 0), (2, 2) and (1, 2), then angle between c) H.P. d) None of these
diagonals is 62. A point equidistant from the lines
4x + 3y + 10 = 0, 5x – 12y + 26 = 0 and
a) b) 7x + 24y – 50 = 0 is
3 2
a) (1, –1) b) (1, 1)
3
c) d) c) (0, 0) d) (0, 1)
2 4
63. Which pair of points lie on the same side of
6.3 Distance of a point from a line 3x – 8y – 7 = 0 ?
55. The points on the line x + y = 4 which lie at a unit a) (0, – 1) and (0, 0)
distance from the line 4x + 3y = 10, are b) (4, – 3) and (0, 1)
a) (3, 1), (–7, 11) b) (3, 1), (7, 11) c) (– 3, – 4) and (1, 2)
c) (– 3, 1), (– 7, 11) d) (1, 3), (–7, 11) d) (– 1, – 1) and (3, 7)
56. If p and p' be the distances from origin to the 64. To which of the following types the straight lines
lines x sec + y cosec = k and represented by 2x + 3y – 1 = 0 and 2x + 3y – 5 = 0
x cos – y sin = k cos 2, then 4p2 + p'2 = belong
a) k b) 2k a) Parallel to each other
2
c) k d) 2k2 b) Perpendicular to each other
57. The distance between two parallel lines c) Inclined at 450 to each other
3x + 4y – 8 = 0 and 3x + 4y – 3 = 0 is given by d) Coincident pair of straight lines
a) 4 b) 5 65. The equations of two lines through (0, a) which
c) 3 d) 1 are at distance 'a' from the point (2a, 2a) are
58. The vertex of an equilateral triangle is (2, –l) and a) y – a = 0 and 4x – 3y – 3a = 0
the equation of its base is x + 2y = 1 . The length b) y – a = 0 and 3x – 4y + 3a = 0
of its sides is c) y – a = 0 and 4x – 3y + 3a = 0
4 2 d) None of these
a) b)
15 15 66. If the equation y = mx + c and x cos + y sin= p
represents the same straight line, then
4 1
c) d)
a) p c 1 m
3 3 2
5
59. The product of the perpendiculars drawn from
b) c p 1 m
2
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DGT MH –CET 11th MATHEMATICS Study Material 11
Straight Line 67
67. The equations of the lines through the point of Competitive Thinking
intersection of the lines x – y + 1 = 0 and
2x – 3y + 5 = 0 and whose distance from the 6.1 Slope of a line, Equation of
a line in different forms
7
point (3, 2) is , is
5 1. The gradient of the line joining the points on the
a) 3x – 4y – 6 = 0 and 4x + 3y + 1 = 0 curve y = x2 + 2x whose abscissa are 1 and 3, is
a) 6 b) 5
b) 3x – 4y + 6 = 0 and 4x – 3y – 1 = 0
c) 4 d) 3
c) 3x – 4y + 6 = 0 and 4x – 3y + 1 = 0
2. The line passing through the points (3, – 4) and
d) None of these
(– 2, 6) and a line passing through (– 3, 6) and
68. Equation of a straight line on which length of (9, – 18)
perpendicular from the origin is four units and
the line makes an angle of 1200 with the X-axis, a) are perpendicular
is b) are parallel
c) make an angle 60° with each other
a) x 3 y 8 0 b) x 3 y 8
d) none of these
c) x 3 y 8 d) x 3y 8 0 3. The lines y = 2x and x = – 2y are
69. In the equation y – y1 = m(x – x1) if m and x1 are a) parallel
fixed and different lines are drawn for different b) perpendicular
values of y1, then
c) equally inclined to axes
a) The lines will pass through a single point
d coincident
b) There will be a set of parallel lines
4. If the line passing through (4, 3) and (2, k) is
c) There will be one line only perpendicular to y = 2x + 3, then k =
d) None of these a) – 1 b) 1
70. The equations of the lines passing through the
c) – 4 d) 4
3 5. The equation of a straight line passing through
point (1, 0) and at a distance from the origin,
2 the point (3, 2) and perpendicular to the line y = x
are is
a) x - y = 5 b) x + y = 5
a) 3x y 3 0, 3x y 3 0
c) x + y = l d) x – y = l
b) 3x y 3 0, 3x y 3 0 6. The equation of the line perpendicular to the line
c) x 3y 3 0, x 3x 3 0 x y
1 and passing through the point at which
a b
d) None of these
it cuts X-axis, is
71. (sin , cos ) and (3, 2) lies on the same side of
the line x + y = 1, then lies between x y a x y b
a) 0 b)
a) (0, /2) b) (0, ) a b b b a a
c) (/4, /2) d) (0, /4)
x y x y a
72. If we reduce 3x + 3y + 7 = 0 to the form c) 0 d)
x cos + y sin = p, then the value of p is b a b a b
7. The equation of a line passing through the point
7 7
a) b) (1, 2) and perpendicular to the line x + y + 1 = 0
2 3 3 is
3 7 7 a) y – x + l = 0 b) y – x – 1 = 0
c) d) c) y – x + 2 = 0 d) y – x – 2 = 0
2 3 2
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DGT MH –CET 11th MATHEMATICS Study Material 12
Straight Line 68
8. The equation of a straight line passing through 17. The equations of the straight line passing through
(– 3, 2) and cutting an intercept equal in magnitude the point (4, 3) and making intercepts on the
but opposite in sign from the axes is given by co-ordinate axes whose sum is – 1, is
a) x – y + 5 = 0 b) x + y – 5 = 0
x y x y
c) x – y – 5 = 0 d) x + y + 5 = 0 a) 1 and 1
2 3 2 1
9. Equation of the line passing through (– 1, 1) and
perpendicular to the line 2x + 3y + 4 = 0 is x y x y
b) 1 and 1
a) 2(y – l) = 3 (x + l) b) 3(y – l) = – 2(x + l) 2 3 2 1
c) y – l = 2 (x + l) d) 3(y – l) = x + l
x y x y
10. The points A (1, 3) and C (5, 1) are the opposite c) 1 and 1
vertices of rectangle. The equation of line passing 2 3 2 1
through other two vertices and of gradient 2, is
x y x y
a) 2x + y – 8 = 0 b) 2x – y – 4 = 0 d) 1 and 1
2 1 2 1
c) 2x – y + 4 = 0 d) 2x + y + 7 = 0
11. A line is such that its segment between the straight 18. The equation to the line bisecting the join of
lines 5x – y – 4 = 0 and 3x + 4y – 4 = 0 is bisected (3, – 4) and (5, 2) and having its intercepts on the
at the point (1,5), then its equation is X-axis and the Y-axis in the ratio 2 : 1 is
a) 83x – 35y + 92 = 0 a) x + y – 3 = 0 b) 2x – y = 9
b) 35x – 83y + 92 = 0 c) x + 2y = 2 d) 2x + y = 7
c) 35x + 35y + 92 = 0 19. If the straight line ax + by + c = 0 always passes
d) None of these through (1, – 2), then a, b, c are in
12. The equation of a line bisecting perpendicularly the a) A.P. b) H.P.
segment joining the points (– 4, 6) and (8, 8) is c) G.P. d) None of these
a) 6x + y – 19 = 0 b) y = 7 20. The inclination of the straight line passing through
c) 6x + 2y – 19 = 0 d) x + 2y – 7 = 0 the point (– 3, 6) and the midpoint of the line joining
13. A (– l, 1), B(5, 3) are opposite vertices of a square the points (4, – 5) and (– 2, 9) is
in jry-plane. The equation of the other diagonal
not passing through (A, B) of the square is given a) b)
by 4 6
a) x – 3y + 4 = 0 b) 2x – y + 3 = 0 3
c) y + 3x – 8 = 0 d) x + 2y – 1 0 c) d)
3 4
14. Equations of diagonals of square formed by lines 21. For what values of a and b, the intercepts cut off
x = 0, y = 0, x = 1 and y = 1 are on the coordinate axes by the line ax + by + 8 = 0
a) y = x,y + x = 1 b) y = x, x + y = 2 are equal in length but opposite in sign to those
1 cut off by the line 2x – 3y + 6 = 0 on the axes ?
c) 2y x, y x d) y = 2x, y + 2x = 1
3
8 8
15. The diagonal passing through origin of a a) a , b 4 b) a , b 4
3 3
quadrilateral formed by x = 0, y = 0, x + y = 1 and
6x + y = 3 is 8 8
a) 3x – 2y = 0 b) 2x – 3y = 0 c) a , b 4 d) a , b 4
3 3
c) 3x + 2y = 0 d) None of these
22. The medians AD and BE of a triangle with
16. Equation of the straight line making equal vertices A(0, b), B(0, 0) and C(a, 0) are
intercepts on the axes and passing through the perpendicular to each other, if
point (2,4) is
a) 4x – y – 4 = 0 b) 2x + y – 8 = 0 a) a 2b b) a 2b
c) x + y – 6 = 0 d) x + 2y – 10 = 0 c) Both (a) and (b) d) None of these
MATHEMATICS - XI OBJECTIVE
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DGT MH –CET 11th MATHEMATICS Study Material 13
Straight Line 69
31. If the line px – qy = r intersects the co-ordinate
3 5
23. If , is the midpoint of line segment axes at (a, 0) and (0, b), then the value of a + b is
2 2
equal to
intercepted by a line between axes, the
equation of the line is q p q p
a) r b) r
a) 5x + 3y + 15 = 0 b) 3x + 5y +15 = 0 pq pq
c) 5x + 3y – 15 = 0 d) 3x + 5y – 15 = 0
p q p q
24. The slope of a line that makes an angle of measure c) r d) r
300 with Y-axis is pq p q
MATHEMATICS - XI OBJECTIVE
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DGT MH –CET 11th MATHEMATICS Study Material 14
Straight Line 70
x y 46. If the lines 4x + 3y = 1, x – y = – 5 and
38. Angle between the lines 1 and
a b 5y + bx = 3 are concurrent, then b equals
x y a) 1 b) 3
1 is
a b c) 6 d) 0
b 2ab 47. If a and b are two arbitrary constants, then the
a) 2 tan 1 b) tan 1 straight line (a – 2b)x + (a + 3b)y + 3a + 4b = 0
a a 2 b2
will pass through
a 2 b2
c) tan 1 d) None of thes a) (– l, – 2) b) (1, 2)
a 2 b2 c) (– 2, – 3) d) (2, 3)
39. The angle between the two lines y – 2x = 9 and 48. If a, b, c are in harmonic progression, then straight
x + 2y = – 1, is
x y 1
a) 600 b) 300 line 0 always passes through a fixed
a b c
c) 900 d) 450
point, that point is
1 1 x y a) (– 1 – 2) b) (– 1, 2)
40. If 0 , then lines 1 and
ab ' ba ' a b c) (l, – 2) d) (l, – l/2)
x y 49. The equation to the sides of a triangle are
1 are x – 3y = 0, 4x + 3y = 5, and 3x + y = 0. The line
b' a '
3x – 4y = 0 passes through
a) Parallel
a) The incentre
b) Inclined at 600 to each other
b) The centroid
c) Perpendicular to each other
c) The circumcentre
d) Inclined at 300 to each other
d) The orthocentre of the triangle
41. Angle between x = 2 and x – 3y = 6 is
50. Two consecutive sides of a parallelogram are
a) b) tan–1 (3) 4x + 5y = 0 and 7x + 2y = 0. If the equation to
1 one diagonal is llx + 7y = 9, then the equation
c) tan 1 d) None of these of the other diagonal is
3
a) x + 2y = 0 b) 2x + y = 0
42. The angle between the lines 2x – y + 3 = 0 and
c) x – y = 0 d) None of these
x + 2y + 3 = 0 is
51. The triangle formed by the lines x + y – 4 = 0,
a) 900 b) 600
3x + y = 4, x + 3y = 4 is
c) 450 d) 300
a) isosceles b) equilateral
43. If the lines ax + by + c = 0, bx + cy + a = 0 and
c) right-angled d) none of these
cx + ay + b = 0 be concurrent, then
52. The lines ax+ by + c = 0, where 3a + 2b + 4c = 0
a) a3 + b3 + c3 + 3abc = 0
are concurrent at the point
b) a3 + b3 + c3 – abc = 0
a) (1/2,3/4) b) (1,3)
c) a3 + b3 + c3 – 3abc = 0
c) (3,1) d) (3/4,1/2)
d) None of these
53. The parallelism condition for two straight lines
44. For what value of ‘a’ the lines x = 3,y = 4 and one of which is specified by the equation
4x – 3y + a = 0 are concurrent ax + by + c = 0, the other being represented
a) 0 b) – 1 parametrically by x = t + , y = t + is given
c) 2 d) 3 by
45. The lines 2x + y – 1 = 0, ax + 3y – 3 = 0 and a) – b = 0, = = c = 0
3x + 2y – 2 = 0 are concurrent for b) a– b = 0, = = 0
a) All a b) a = 4 only c) a + b = 0
c) 1 a 3 d) a > 0 only d) a = b = 0
MATHEMATICS - XI OBJECTIVE
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DGT MH –CET 11th MATHEMATICS Study Material 15
Straight Line 71
54. The equation of the line which passes through 61. The straight lines x + y = 0, 5x + y = 4 and
the point (3, – 2) and are inclined at 600 to the x + 5y = 4 form
line 3x y 1, is a) an isosceles triangle
b) an equilateral triangle
a) y 2 0, 3x y 2 3 3 0 c) a scalene triangle
b) x 2 0, 3x y 2 3 3 0 d) a right angled triangle
6.3 Distance of a point from a line
c) 3x y 2 3 3 0
62. The points on the X-axis whose perpendicular
d) x 3y 2 3 3 0 x y
55. The point (– 4, 5) is the vertex of a square and
distance from the line 1 is a, are
a b
one of its diagonals is 7x – y + 8 = 0. The equation
of the other diagonal is a
a) (b a 2 b 2 ), 0
a) 7x – y + 23 = 0 b) 7y + x = 30 b
c) 7y + x = 3l d) x – 7y = 30
b
b) (b a b ), 0
2 2
56. The line passing through the point of intersection
a
of x + y = 2, x – y = 0 and is parallel to
x + 2y = 5 is a
c) (a a 2 b 2 ), 0
a) x + 2y = 1 b) x + 2y = 2 b
c) x + 2y = 4 d) x + 2y = 3 d) None of these
57. The line parallel to the X-axis and passing through 63. The distance between the lines 3x + 4y = 9 and
the intersection of the lines ax + 2by + 3b = 0 and 6x + 8y = 15 is
bx - 2ay – 3a = 0, where (a, b) (0, 0) is
3 3
a) above the X-axis at a distance of 3/2 from it a) b)
2 10
b) above the X-axis at a distance of 2/3 from it
c) 6 d) None of these
c) below the X-axis at a distance of 3/2 from it .
64. The length of the perpendicular drawn from origin
d) below the X-axis at a distance of 2/3 from it
The angle between the lines x y
58. upon the straight line 1 is
x cos + y sin = a and x sin – y cos = a is 3 4
a) – b) + – 2 1
a) 2 b) 3
5 5
c) d)
2 2 2 2
c) 4 d) 3
59. A line passes through the point of intersection of 5 5
the lines 3x + y + 1 = 0 and 2x – y + 3 = 0 and 65. Two points A and B have co-ordinates (1, 1) and
makes equal intercepts with axes. The equation (3, – 2) respectively. The co-ordinates of a point
of the line is distant 85 from B on the line through
a) 5x + 5y – 3 = 0 b) x + 5y – 3 = 0 B perpendicular to AB are
c) 5x – y – 3 = 0 d) 5x + 5y + 3 = 0 a) (4, 7) b) (7, 4)
60. The length of the straight line x – 3y = 1 c) (5, 7) d) (– 5, – 3)
intercepted by the hyperbola x2 – 4y2 = 1, is
66. The equation of the base of an equilateral triangle
6 is x + y = 2 and the vertex is (2, 1). The length of
a) 10 units b) units the side of the triangle is
5
6 6 a) 3/ 2 b) 2
c) units d) 10 units
10 5 c) 2/3 d) None of these
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DGT MH –CET 11th MATHEMATICS Study Material 16
Straight Line 72
67. Let be the distance between the lines 73. The equation of one of the lines parallel to
– x + y = 2 and x – y = 2 and be the distance 4x – 3y = 5 and at a unit distance from the point
between the lines 4x – 3y = 5 and 6y – 8x = 1, (– 1, – 4) is
then a) 3x + 4y – 3 = 0 b) 3x + 4y + 3 = 0
a) 20 2 11 b) 20 2 11 c) 4x – 3y + 3 = 0 d) 4x – 3y – 3 = 0
74. The length of the perpendicular from the origin
c) 11 2 20 d) None of these
x cos y cos
68. Choose the correct statement which describe the
on the line 1 0 is
b a
position of .the point (– 6, 2) relative to straight
75. The number of points on the line x + y = 4 which
lines 2x + 3y – 4 = 0 and 6x + 9y + 8= 0 ?
are unit distance apart from the line 2x + 2y = 5
a) Below both the lines is
b) Above both the lines a) 0 b) 1
c) In between the lines c) 2 d) Infinity
d) None of these 76. The distance between the parallel lines
69. If the equation of base of an equilateral triangle y = x + a, y = x + b is
is 2x – y = 1 and the vertex is (– 1, 2), then the
length of the side of the triangle is |a b|
a) b) |a – b|
2
20 2
a) b)
3 15 |a b|
c) |a + b| d)
2
8 15
c) d) 77. The equation of straight line equally inclined to
15 2 the axis and equidistant from the points (1, – 2)
70. The vertices of a triangle are (2, 1), (5, 2) and and (3, 4) is ax + by + c = 0, where
(4, 4). The lengths of the perpendicular from these a) a = l, b = l, c = l b) a = l, b = l, c = – 1
vertices on the opposite sides are c) a = 1, b = l, c = 2 d) None of these
7 7 7 7 7 7 78. A straight line passes through the points (5, 0)
a) , , b) , , and (0, 3). The length of perpendicular from the
5 13 6 6 8 10
point (4, 4) on the line is
7 7 7 7 7 7
c) , , d) , , 15 17
5 8 15 5 13 10 a) b)
34 2
71. The vertices of a triangle OBC are (0, 0),
(– 3, – 1) and (– 1, – 3) respectively. Then the 17 17
c) d)
1 2 2
equation of line parallel to BC which is atunit
2
distant from origin and cuts OB and OC, is
a) 2x 2y 2 0 b) 2x 2y 2 0
c) 2x 2y 2 0 d) None of these
x y
72. The line L given by 1 passes through
5 b
the point (13, 32). The line K is parallel to K and
x y
the equation 1 . Then, the distance
c 3
between L and K is
MATHEMATICS - XI OBJECTIVE
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DGT MH –CET 11th MATHEMATICS Study Material 17
Straight Line 73
Evaluation Test 7. A square of side V lies above the X-axis and has
one vertex at the origin. The side passing through
1. If the line y = 7x – 25 meets the circle x2 +y2 = 25
the origin makes an angle (0 < < /4) with
at the points A, B, then the distance between A and
the positive direction of X-axis. The equation
B is
of its diagonal not passing through the origin is
a) 10 b) 10 a) y(cos + sin ) + x(cos – sin ) = a
c) 5 2 d) 5 b) y(cos – sin ) – x(sin – cos ) = a
2. If f() = x cos + y sin – p(), then the lines c) y(cos + sin ) + x(sin – cos ) = a
f() = 0 and f() = 0 are perpendicular to each d) y(cos + sin ) + x(sin + cos ) = a
other, if 8. A line L passes through the points (1, 1) and
1
a) = b) (2, 0) and another line L’ passes through , 0
2 2
and perpendicular to L. Then the area of the
c) | | d) none of these
2 triangle formed by the lines L, L' and Y-axis is
3. The locus of the orthocentre of the triangle formed 15 25
by the lines (1 + p)x – py + p(l + p) = 0, a) b)
8 4
(1 + q)x – qy + q(l + q) = 0 and y = 0, where
p q, is 25 25
c) d)
a) a hyperbola b) a parabola 8 16
c) an ellipse d) a straight line 9. The number of integer values of m, for which the
4. If the straight line ax + by + c = 0 make a triangle x-co-ordinate of the point of intersection of the
of constant area with coordinate axes, then lines 3x + 4y = 9 and y = mx + 1 is also an integer,
is
a) a, b, c are in G.P. b) a, c, b are in G.P.
a) 2 b) 0
c) c, a, b are in G.P. d) none of these
c) 4 d) 1
10. If straight lines ax + by + p = 0 and x cos + y
5. Let 0 be a fixed angle.
2
If P =(cos, sin) and Q =(cos(– ), sin( – )), sin – p = 0 are inclined at an angle and
4
then Q is obtained from P by
meet the straight line x sin – y cos = 0 at the
a) clockwise rotation around origin through an same point, then the value of a2 + b2 is equal to
angle
a) 1 b) 2
b) anti-clockwise rotation around origin through
c) 3 d) 4
an angle
11. A line Ax + y =1 passes through the point
c) reflection in the line through origin with slope
A(2, –7) meets the line BC whose equation is
tan
3x – 4y + 1 = 0 at the point B. The equation to
d) reflection in the line through origin with the line AC so that AB = AC is
tan
slope a) 52x + 89y + 519 = 0
2
b) 52x + 89y – 519 = 0
6. Let P(– 1, 0), Q(0, 0) and R (3, 3 3) be three c) 89x + 52y + 519 = 0
points. The equation of the bisector of the angle d) 89x + 52y – 519 = 0
PQR is
3
a) xy0 b) x 3y 0
2
3
c) 3x y 0 d) x y0
2
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DGT MH –CET 11th MATHEMATICS Study Material 18
Straight Line 74
Answer Key
MATHEMATICS - XI OBJECTIVE
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DGT MH –CET 11th MATHEMATICS Study Material 19
DGT Group - Tuitions (Feed Concepts) XIth – XIIth | JEE | CET | NEET | Call : 9920154035 / 8169861448
20
DGT Group - Tuitions (Feed Concepts) XIth – XIIth | JEE | CET | NEET | Call : 9920154035 / 8169861448
21
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DGT Group - Tuitions (Feed Concepts) XIth – XIIth | JEE | CET | NEET | Call : 9920154035 / 8169861448
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DGT Group - Tuitions (Feed Concepts) XIth – XIIth | JEE | CET | NEET | Call : 9920154035 / 8169861448
25
DGT Group - Tuitions (Feed Concepts) XIth – XIIth | JEE | CET | NEET | Call : 9920154035 / 8169861448
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DGT Group - Tuitions (Feed Concepts) XIth – XIIth | JEE | CET | NEET | Call : 9920154035 / 8169861448
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DGT Group - Tuitions (Feed Concepts) XIth – XIIth | JEE | CET | NEET | Call : 9920154035 / 8169861448
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