Line and Lines (Solved MCQs and Notes)

Notes 55 Pages

Joint Entrance Examination (Main)

Mathematics

Contributed by

Nikita Narasimhan

37
Unit-11 Lines
1. The equation of line equidistant from the points A(1, –2) and B(3,4) and making congruent
angles with the coordinate axes is . . .
(a) x +y = 1 (b) y – x + l = 0 (c)y – x – 1 = 0 (d) y – x = 2
2. The equation of line passing through the point (–5,4) and making the intercept of length
2
5
between the lines x + 2y – 1 = 0 and x + 2y + 1 = 0 is . . .
(a) 2x – y + 4 = 0 (b) 2x – y –14 = 0 (c) 2x – y + 14 = 0 (d) None of these
3. The equation of line containing the angle bisector of the lines 3x – 4y – 2 = 0 and 5x –
12y + 2 = 0 is . . .
(a) 7x + 4y – 18 = 0 (b) 4x – 7y – 1 = 0 (c) 4x – 7y + 1 = 0 (d) None of these
4. The equation of line passing through the point of intersection of the lines
3x – 2y = 0 and 5x + y – 2 = 0 and making the angle of measure tan
–1
(–5) with the positive
direction of x – axis is . . .
(a) 3x – 2y = 0 (b) 5x + y – 2 = 0 (c) 5x + y = 0 (d) 3x + 2y + 1 = 0
5. If for a + b + c  0, the lines ax + by + c = 0, bx + cy + a = 0 and cx + ay + b = 0 are
concurrent, then . . .
(a) ab + be + ca = 0 (b)
b c
+ + 1
b c
a
a

(c) a = b (d) a = b = c
6. The equation of line passing through the point (1,2) and making the intercept of length 3
units between the lines 3x + 4y = 24 and 3x + 4y = 12, is . . .
(a) 7x – 24y + 41 = 0 (b) 7x + 24y = 55 (c) 24x – 7y = 10 (d) 24x + 7y – 38 = 0
7. If (a, a
2
) lies inside the angle between the lines y =
2
x
, x > 0 and y = 3x, x > 0,
then a  . . .
(a)
 
1
2
3,  
(b) (3,

) (c)
 
1
2
, 3
(d)
 
1
2
0,
8. If P(–1,0), Q(0,0) and
R(3, 3 3)
, then the equation of bisector of PQR is . . .
(a)
3
2
y 0x  
(b)
3
2
y 0x  
(c)
3 y 0x  
(d)
3y 0x  
Page 1
38
9. If the non zero numbers a, b,c are in harmonic progression, then the line
y 1
0
b c
x
a
  
passes through the point . . .
(a) (1,–2) (b) (–1,–2) (c) (–1,2) (d)


1
2
1,
10. A line passing through 0(0,0) intersect the parallel lines 4x + 2y = 0 and 2x + y + 6 = 0 at
P and Q respectively, then in what ratio does 0 divide
PQ
from P ?
(a) 1 : 2 (b) 3 : 4 (c) 2 : 1 (d) 4 : 3
11. The points on the line 3x – 2y – 2 = 0, which are 3 units away from the line
3x + 4y – 8 = 0 are . . .
(a)
 
1
3
(3, 3), 3,  
(b)
 
 
7 3
1
2 3 2
3, , , 
(c)
 
 
7
1
2 3
,3 , ,3
(d) (3,1),(1,3)
12. If A(1, –2), 5(–8,3), A–P–B and 3 AP = 7AB, then P = . . .
(a)
 
41
3
22, 
(b)
 
41
3
22,
(c) not possible (d) None of these
13. For the collinear points P – A – B, AP = 4AB, then P divides
AB
from A in the ratio.....
(a) 4 : 5 (b) – 4 : 5 (c) –5 : 4 (d) –1 : 4
14. If the length of perpendicular drawn from (5,0) on kx + 4y = 20 is 1, then k = . . .
(a)
16
3
3, (b)
16
3
3,  (c)
16
3
3,  (d)
16
3
3,  
15. If the lengths of perpendicular drawn from the origin to the lines xcos

– ysin

=
sin2a

and xsin

+ ycos

= cos2

are p and q respectively, then p
2
+ q
2
= . . .
(a) 4 (b) 3 (c) 2 (d) 1
16. The points onY – axis at a distance 4 units from the line x + 4y = 12 are . . .
(a) (3 14, 0) (3 14, 0)  (b)
( 3 17, 0) (3 17, 0)  
(c)
(0, 3 17)
(d) (0, 3 17) (0, 3 17)   
17. A base of a triangle is along the line x = b and its length is 2b. If the area of triangle is b
2
,
then the vertex of a triangle lies on the line . . .
(a) x =–b (b) x = 0 (c) x =
b
2
(d )x = b
18. Shifting origin at which point the transformed form of x
2
+ y
2
– 4x – 8y – 85 = 0 would
be x
2
+ y
2
= k?
(a) (2,4) (b) (–2, –4) (c) (2, –4) (d) (–2,4)
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39
19. A(1,0) and B(–1,0), then the locus of points satisfying AQ – BQ = ±1 is . . .
(a) 12x
2
+ 4y
2
= 3 (b) 12x
2
– 4y
2
= 3 (c) 12x
2
– 4y
2
= –3 (d) 12x
2
+ 4y
2
= –3
20. A rod having length 2c moves along two perpendicular lines, then the locus of the mid point
of the rod is . . .
(a) x
2
– y
2
= c
2
(b) x
2
+ y
2
= c
2
(c) x
2
+ y
2
= 2c
2
(d) None of these
21. Consider a square PQR having the length of side a, where O(0,0). The sides
OP
and
OR
are along the positive X – axis and Y – axis respectively. If A and B are the mid points
of
PQ
and
QR
respectively, then the angle between
OA
and
OB
would be... . .
(a) cos
–1
3
5
(b) tan
–1 4
3
(c) cos
–1
3
4
(d) sin
–1
3
5
22.
3 y 2x  
is the equation of line containing one of the sides of an equilateral triangle
and if (0,–1) is one of the vertices, then the length of the side of the triangle is . . .
(a)
3
(b)
2 3
(c)
3
2
(d)
2
3
23. If the point
 
t t
2 2
1 , 2 
lies between the two parallel lines x + 2y = 1 and
2x + 4y = 15, then the range of t is . . .
(a)
5
6 2
0 t 
(b)
4 2
3
t 0  
(c)
4 2 5 2
3 6
t  
(d) None of these
24. If two perpendicular lines passing through origin intersect the line
y
b
1, 0, b 0
x
a
a   
at
A and B, then
1 1
2 2
OA OB
.......... 
(a)
2 2
1 1
b
a

(b)
2 2
b
b
a
a 
(c)
2 2
2 2
b
b
a
a

(d) None of these
25. The equation of a line at a distance
5
units from the origin and the ratio of the intercepts
on the axes is 1 : 2, is . . .
(a) 2x + y + 5 = 0 (b) 2x + y + 5 = 0 (c) x – 2y + 5 = 0 (d) None of these
26. For any values of p and q, the line (p + 2q)x + (p – 3q)y – p – q passes through which
fixed point ?
(a)
 
3 5
2 2
,
(b)
 
2 2
5 5
,
(c)
 
3 3
5 5
,
(d)
 
3
2
5 5
,
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40
27. If A(x
1
,y
1
), B(x
2
,y
2
) and P(tx
2
+ (1 – t)x
1
, ty
2
+ (1 – t)y
1
) where t < 0, then P divides
AB
from A in the ratio . . .
(a) 1 – t (b)
t 1
t

(c)
t
1 t
(d) t – 1
28. A(1,2), B(5,7) and P(x,y)
AB

, then y – x – 1 is . . .
(a) < 0 (b) > 0 (c) = 0 (d) –3
29. A(2,3), B(4,7) and P(x,y)
AB
, then the maximum value of 3x + y is . . .
(a) 19 (b) 9 (c) –19 (d) –9
30. A(– 2,5), 5(6,2), then
AB AB .......... 

(a) {(8t – 2, 5 – 3t / t < 0) (b) {(8t – 2, 5 – 3t) / 0 < t < 1}
(c) {(8t – 2, 5 – 3t) / t R – [0, 1]} (d) {(8t – 2, 5 – 3t) / t > 1}
31. The p –  form of the line
3y 4 0x   
is
(a)
π π
2
6 6
xcos ysin 
(b)
π π
2
3 3
xcos ysin 
(c)
π π
2
3 3
xcos ysin
   
   
   
   
(d)
π π
2
6 6
xcos ysin
   
   
   
   
32. The length of side of an equilateral triangle is a. There is circle inscribed in a triangle.
What is the area of a square inscribed in a circle ?
(a)
2
3
a
(b)
2
6
a
(c)
3
2
a
(d)
2
2
a
33. If the lines x + 2ay + a – 0, x + 3by + 3 = 0 and x + 4cy + c = 0 are concurrent, then a,
b, c are in . . .
(a) A.P. (b) H.P. (c) G.P. (d) A.G.P
34. The foot of perpendicular drawn from (2,3) to the line 4x – 5y – 34 = 0 is . . .
(a) (6,–2) (b)
 
246 82
41 41
,
(c) (–6,2) (d) None of these
35. The equation of a line passing through (4,3) and the sum of whose intercepts is –1 is.....
(a)
y y
2 3 2 1
1, 1
x x
   
(b)
y y
2 3 2 1
1, 1
x x

    
(c)
y y
2 3 2 1
1, 1
x x

     
(d)
y y
2 3 2 1
1, 1
x x

   
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41
36. A line intersects X – axis and Y – axis at A and B respectively. If AB = 15 and AB

makes a
triangle of area 54 units with coordinate axes, then the equation of
AB

is . . .
(a) 4x ± 3y = 36 or 3x ± 4y = 36 (b) 4x ± 3y = 24 or 3x ± 4y = 24
(c) –4x ± 3y = 24 or – 3x ± 4y – 24 (d) –4x ± 3y = 12 or – 3x ± 4y – 12
37. The angle between the lines xcos85° + ysin85° = 1 and xcos40° + ysin40° = 2 is :
(a) 90° (b) 80° (c) 125° (d) 45°
38. If a
1
, a
2
, a
3
and b
1
, b
2
, b
3
are in geometric progression and their common ratios are equal,
then the points
A(a
1
, b
1
), B(a
2
,b
2
) and C(a
3
,b
3
) . . .
(a) lie on the same line (b) lie on a circle (c) lie on an ellipse (d) None of these
39. The image of the point (4, –13) in the line 5x + y + 6 = 0 is . . .
(a) (1,2) (b) (3,4) (c) (–4,13) (d) (–1, –14)
40. If the lines x + (a – l)y + 1 = 0 and 2x + a
2
y –1 = 0 are perpendicular then . . .
(a) | a | = 2 (b) 0 < a < 1 (c) –1 < a < 1 (d) a = –1
41. If x + 3y – 4 = 0 and 6x – 2y – 7 = 0 are the lines containing the diagonals of a
parallelogram PQRS, then parallelogram PQRS is . . .
(a) rectangle (b) square (c) cyclic quadrilateral (d) rhombus
42. For a + b + c = 0, the line 3ax + 4by + c = 0 passes through the fixed point . . .
(a)
 
1 1
3 4
, 
(b)
 
1 1
3 4
, 
(c)
 
1 1
3 4
,
(d)
 
1 1
3 4
,  
43. If 3l + 2m + 6n = 0, then the family of lines lx + my + n = 0 passes through the fixed
point . . .
(a) (2,3) (b) (3,2) (c)
 
1 1
2 3
,
(d)
 
1 1
3 2
,
44. If the lines x + y + r = 0 and

x – 5y = 5 are identical then

+ r = . . ,
(a) –4 (b) 4 (c) 1 (d) –1
45. If the x – coordinate of the point of intersection of the lines 3x + 4y = 9 and y = mx + 1
is an integer, then the integer value of m is . . .
(a) 2 (b)0 (c) 4 (d) 1
46. If (4,5) is the foot of perpendicular on the line l, then the equation of the line l would
be . . .
(a) 4x + 5y + 41 = 0 (b) 4x – 5y + 9 = 0 (c) 4x + 5y – 41 = 0 (d) None of these
47. The y – intercept of the line y + y
1
= m(x – x
1
) is . . .
(a) – (y
1
+ mx
1
) (b) y
1
– mx
1
(c)
1 1
y m
m
x

(d) None of these
Page 5
42
48. The locus of mid points of the segment intercepted between the axes by the line
xseca + ytana = p is . . .
(a)
2 2
2 2
p p
4 4y
1
x
 
(b)
2
2
2 2
y
p p
4
x
 
(c)
2 2
2 2
1
p p
x y
 
(d)
2 2
2 2
1
p p
4x 4y
 
49. If the y – intercept of the perpendicular bisector of the segment obtained by joining
P(l,4) and Q(k, 3) is –4 then k = . . .
(a) 1 (b) 2 (c) –2 (d) –4
50. The y – intercept of the line passing through the point (2,2) and perpendicular to the line
3x + y – 3 = 0 is . . .
(a)
3
4
(b)
4
3
(c)
4
3
 (d)
3
4

51. The line parallel to the X – axis and passing through the intersection of the lines
ax + 2by + 3b = 0 and bx – 2ay – 3a = 0 where (a, b)  (0,0) is :
(A) above the X – axis at a distance of
2
3
from it
(B) above the X – axis at a distance of
3
2
from it
(C) below the X – axis at a distance of
2
3
from it
(D) below the X – axis at a distance of
3
2
from it
52. A square of side a lies above the x – axis and has one vertex at the origin. The side passing
through the origin makes an angle a
 

 

  
with the positive direction of x – axis.
The eq. of its diagonal not passing through the origin is :
(A) y(cos

+ sin

) + x(sin

– cos

) = a (B) y(cos

+ sin

) + x(sin

+ cos

) = a
(C) y(cos

+ sin

) + x(cos

– sin

) = a (D) y(cos

– sin

) – x(sin

– cos

) = a
53. If P and Q divides
AB
from A in the ratios

and –

, then A divides
PQ
from p in the
ratio . . . . . . .
(a)
1



+1
(b)
1
1




(c)
2
2




(d)
2
2




54. The nearest point on the line x – 3y + 25 = 0 from the origin is . . .
(a) (–4,5) (b) (–4,3) (c) (4,3) (d) None of these
55. If the slope of a curve is constant, then the graph of a curve in the plane is . . .
(a) line (b) parabola (c) hyperbola (d) none of these
56. If 5x + 12y + 13 = 0 is transformed into xcos

+ ysin

= p, then

= ?


(a) cos
–1
 
5
13

(b) sin
–1
 
12
13

(c) tan
–1
 
12
5


(d) tan
–1
 
12
5
Page 6
43
57.
I f
P(–1,0), Q(0,0) and
R(3,3 3)
are given points, then the equation of the bisector of
PQR is . . .
(a)
3
2
yx   
(b)
3
2
0x y 
(c)
3 y 0x  
(d)
3y 0x  
58. For the line y – y
1
= m(x – x
t
), m and x
1
are fixed values, if different lines are drawn
according to the different value of y
1;
then all such lines would be . . .
(a) all lines intersect the line x = x
1
(b) all lines pass through one fixed point
(c) all lines are parallel to the line y = x
1
(d) all lines will be the set of perpendicular lines
59. If the length of perpendicular drawn from origin to a line is 10 and
5
6


  then the
equation of line would be . . .
(a)
3 yx   
(b)
3 yx   
(c)
3 y 20 0x   
(d)
3 y 20 0x   
60. Find the equation of line making a triangle of area
50
3
units with two axes and on which a
perpendicular from origin makes an angle


with positive direction of x – axis.
(a)
3yx  
(b) x – y = 10 (c)
3 y 5x    
(d)
3 y 10x    
61. If 2x + 2y – 5 = 0 is the equation of the line containing one of the sides of an equilateral
triangle and (1,2) is one vertex, then find the equations of the lines containing the other
two sides.
(a)
y (2 3) 3, y 3) 3x x      
(b)
y (2 3) 3, y 3) 3x x      
(c)
y (2 3) 3, y 3) 3x x      
(d)
y (2 3) 3, y 3) 3x x      
62. Find the equation of line passing through the point ( 3, 1) and at a distance
2
units
from the origin.
(a)
( 3 1) ( 3 1) or ( 3 1) ( 3 1)yx y x         
(b)
( 3 1) ( 3 1) or ( 3 1) ( 3 1)yx y x         
(c)
( 3 1) ( 3 1) or ( 3 1) ( 3 1)yx y x         
(d)
( 3 1) ( 3 1) or ( 3 1) ( 3 1)yx y x         
Page 7
44
63. If (3,–2) and (–2,3) are two vertices and (6,–1) is the orthocentre of a triangle, then the
third vertex would be . . .
(a) (1,6) (b) (–1,6) (c) (1, –6) (d) none of these
64. The circumcentre of the triangle formed by the lines x + y = 0, x – y = 0 and x – 7 = 0 is . .
(a) (7,0) (b) (3.5,0) (c) (0,7) (d) (3.5,3.5)
65. If
1 1 1
b c
, ,
a
are in arithmetic sequence, then the line
y
1
b c
0
x
a
  
passes through the fixed
point . . .
(a) (–1,–2) (b) (–1,2) (c)
 
1
2
1, 
(d) (1,–2)
66. Find the slope of the line passing through the point (1,2) and the point of intersection of
this line with the line x + y + 3 = 0 is at a distance
3 2
units from (1,2).
(a)
1
3
(b) 1 (c)
3
(d)
3 1
2

67. The angle between the lines x = 3 and
3 5x y   
is . . .
(a)


(b)
3

(c)
4

(d)
2

68. The angle between the lines y = e and
3 y 5x    
is . . .
(a)



(b)
5


(c)


(d)
3

69.
T he angl e betw een the lines
{(x, 0)/x

R} and {(0,y)/ y

R} is . . .
(a)
2

(b)
2


(c) 0 (d) 
70. If the point
 
t t
2 2
1 , 2 
lies between the two parallel lines x + 2y = 1 and 2x + 4y = 15,
then the range of t is . . .
(a)
5
6 2
0 t 
(b)
4 2
3
t 0   (c)
4 2 5 2
3 6
t   (d) None of these
71. If 2x + 3y = 8 is perpendicular to the line (x + y + 1) +

(2x – y – 1) = 0, then

= ?
(a) –5 (b)
3
2
(c) 5 (d) 0
72. If the line (a + l)x + (a
2
— a — 2)y + a = 0 is parallel to Y – axis, then a = . . .
(a) –1 (b) 2 (c) 3 (d) 1
Page 8
45
73. The equation of a straight line passing through the point (–5, 4) and which cut off an
intercept of
2
unit between the lines x + y + 1 = 0 and x + y – 1 = 0 is
(a) x – 2y + 13 = 0 (b) 2x – y + 14 = 0 (c) x – y + 9 = 0 (d) x – y + 10 = 0
74. If P(1, 2), Q(4, 6), R(5, 7) and S(a, b) are the vertices of a parallelogram PQRS then
(a) a = 2, b = 4 (b) a = 3, b = 4 (c) a = 2, b = 3 (d) a = 2, b = 5
75. The sum of squares of intercepts on the axes cut off by the tangents to the curve
2 2 2
3 3 3
yx a 
(a > 0) at


8 8
,
a a
is 2. Thus a has the value.
(a) 1 (b) 2 (c) 4 (d) 8
76. If two vertices of a trinangle eare (5, –1) and (–2, 3) and if its orthocentre lies at the origin
then the cooridnates of the third vertex are
(a) (4, 7) (b) (–4, –7) (c) (2, –3) (d) (5, –1)
77. Line ax + by + p = 0 makes angle


which
+
, R .x cos ysin p p
 
  
If these lines and
the line
0x sin y cos
 
 
are concurrent then
(a) a
2
+ b
2
= 1 (b) a
2
+ b
2
= 2 (c) 2(a
2
+ b
2
) = 1 (d) a
2
– b
2
= 2
78. A straight line passess through a point A(1, 2) and makes an angle 60
0
with the x–axis. This
line intersect the line x + y = 6 at P. Then AP will be
(a)
3( 3 1)
(b)
3( 3 1)
(c)
( 3 1)
(d)
3 3
79. The image of origin in the line x + 4y = 1 is
(a)


8
2
17 17
,

(b)


8
2
17 17
,  
(c)


8
2
17 17
, 
(d)


8
2
17 17
,
80. Orthocentre of triangle with vertices (0, 0), (3, 4) and (4, 0) is
(a)


5
4
3,
(b) (3, 12) (c)


3
4
3,
(d) (3, 9)
81. The equation of three sides of triangle are x = 2, y + 1 = 0 and x + 2y = 4. The coordinates
of the circumcentre of the triangle is
(a) (4, 0) (b) (2, –1) (c) (0, 4) (d) (–1, 2)
82. If a, b, c are in A.P. then ax + by + c = 0 represents
(a) a single line (b) a family of concurrent lienes
(c) a family of parallel lines (d) a family of circle
83. A(4, 0), B(0, 3), C(6, 1) be vertices of triangle ABC. Slope of bisector of angle C will be
(A)
3 2 7
(b)
5 2 7
(c)
6 2 7
(d) none
Page 9
46
84. The locus of the variable point whose distance from (–2, 0) is
2
3
times its distance from
the line
9
2
x  
is
(a) ellipse (b) parabola (c) circle (d) hyperbola
85. The line 3x – 4y + 7 = 0 is rotated through an angle


in the clockwise direction about
the point (–1, 1). The equation of the line in its new position is
(a) 7y + x – 6 = 0 (b) 7y – x – 6 = 0 (c) 7y + x + 6 = 0 (d) 7y – x + 6 = 0
86. The area of the triangle formed by the point (a, a
2
), (b, b
2
), (c, c
2
) is ..... (a, b, c are three
consecutive odd integers)
(a)
1
2
(a – b) (b – c) sq unit (b) 8 sq unit
(c) 16 sq unit (d)
1
2
(a – b) (b – c) (a + b + c) sq unit
87. The straight line 7x - 2y + 10 = 0 and 7x + 2y – 10 = 0 forms an isosceles triangle with
the line y = 2. Area of the triangle is equal to :
(a)
15
7
sq unit (b)
10
7
sq unit (c)
18
7
sq unit (d)
10
13
sq unit
88. In triangle ABC, equation of right bisectors of the sides
AB
and
AC
are x + y = 0 and
y – x = 0 respectively. If A = (5, 7) then equation of side BC is
(a) 7y = 5x (b) 5x = y (c) 5y = 7x (d) 5y = x
89. The equations of the two lines each passing through (5, 6) and each making an acute angle
of 45
0
with the line 2x – y + 1 = 0 is
(a) 3x + y – 21 = 0, x – 3y + 13 = 0 (b) 3x + y + 21 =0, x + 3y + 13 = 0
(c) y = 2x, y = 3x (d) 3x + y – 21 = 0, x – 3y – 13 = 0
90. If the equation of base of an equilateral triangle is 2x – y = 1 and the vertex is (–1, 2),
then the length of the side of the triangle is
(a)
20
3
(b)
2
15
(c)
8
15
(d)
15
2
91. Four points (x
1
, y
1
), (x
2
, y
2
), (x
3
, y
3
) and (x
4
, y
4
) are such that
 
4
2 2
1
i i
i
x y

 

2(x
1
x
3
+ x
2
x
4
+ y
1
y
2
+ y
3
y
4
). Then these points are vertices of
(a) parallellogram (b) Rectangle (c) Square (d) Rhombus
Page 10

Line and Lines (Solved MCQs and Notes)

Line and Lines (Solved MCQs and Notes)

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