circle st.line

JEE - ADVANCE
MATHEMATICS
ADVANCE PROBLEMS ON
CIRCLE & STRAIGHT LINE
Time Limit : 5 Sitting of 90 Minutes and 1 of 2 hrs.
(Only one is correct)
Q.1
Coordinates of the centre of the circle which bisects the circumferences of the circles
x2 + y2 = 1 ; x2 + y2 + 2x – 3 = 0 and x2 + y2 + 2y – 3 = 0 is
(A) (–1, –1)
(B) (3, 3)
(C) (2, 2)
(D) (– 2, – 2)
Q.2
One side of a square is inclined at an acute angle  with the positive x-axis, and one of its extremities is
at the origin. If the remaining three vertices of the square lie above the x-axis and the side of a square is
4, then the equation of the diagonal of the square which is not passing through the origin is
(A) (cos  + sin ) x + (cos  – sin ) y = 4 (B) (cos  + sin ) x – (cos  – sin ) y = 4
(C) (cos  – sin ) x + (cos  + sin ) y = 4 (D) (cos  – sin ) x – (cos  + sin ) y = 4 cos 2
Q.3
The line 2x – y + 1 = 0 is tangent to the circle at the point (2, 5) and the centre of the circles lies on
x – 2y = 4. The radius of the circle is
(A) 3 5
Q.4
(B) 5 3
(C) 2 5
(D) 5 2
Given the family of lines, a (2x + y + 4) + b (x  2y  3) = 0 . Among the lines of the family, the number
of lines situated at a distance of 10 from the point M (2,  3) is :
(A) 0
(B) 1
(C) 2
(D) 
Q.5
The co-ordinate of the point on the circle x² + y²  12x  4y + 30 = 0, which is farthest from the
origin are :
(A) (9 , 3)
(B) (8 , 5)
(C) (12 , 4)
(D) none
Q.6
The area of triangle formed by the lines x + y – 3 = 0 , x – 3y + 9 = 0 and 3x – 2y + 1= 0
(A)
16
sq. units
7
(B)
10
sq. units
7
(C) 4 sq. units
(D) 9 sq. units
Q.7
The number of common tangent(s) to the circles x² + y² + 2x + 8y  23 = 0 and
x² + y²  4x  10y + 19 = 0 is :
(A) 1
(B) 2
(C) 3
(D) 4
Q.8
The four points whose coordinates are (2, 1), (1, 4), (4, 5), (5, 2) form :
(A) a rectangle which is not a square
(B) a trapezium which is not a parallelogram
(C) a square
(D) a rhombus which is not a square.
Q.9
From the point A (0 , 3) on the circle x² + 4x + (y  3)² = 0 a chord AB is drawn & extended to
a point M such that AM = 2 AB. The equation of the locus of M is :
(A) x² + 8x + y² = 0
(B) x² + 8x + (y  3)² = 0
(C) (x  3)² + 8x + y² = 0
(D) x² + 8x + 8y² = 0
Q.10
A ray of light passing through the point A (1, 2) is reflected at a point B on the x  axis and then passes
through (5, 3) . Then the equation of AB is :
(A) 5x + 4y = 13
(B) 5x  4y =  3
(C) 4x + 5y = 14
(D) 4x  5y =  6
Q.11
Two circles of radii 4 cms & 1 cm touch each other externally and  is the angle contained by their
Circle & Straight line
direct common tangents. Then sin  =
(A)
Q.12
24
25
12
25
(C)
3
4
(D) none
If A & B are the points ( 3, 4) and (2, 1), then the coordinates of the point C on AB produced such
that AC = 2 BC are :
(A) (2, 4)
Q.13
(B)
(C) (7, 2)
(B) (3, 7)
 1 5
 2 2
(D)   , 
The locus of the mid points of the chords of the circle x2 + y2  ax  by = 0 which subtend a right angle
 a b
 2 2
at  ,  is :
(B) ax + by = a2 + b2
(A) ax + by = 0
(C)
x2 + y2  ax  by +
a 2 b 2
=0
8
(D)
x2 + y2  ax  by 
a 2 b 2
=0
8
Q.14
The base BC of a triangle ABC is bisected at the point (p, q) and the equation to the side AB & AC are
px + qy = 1 & qx + py = 1 . The equation of the median through A is :
(A) (p  2q) x + (q  2p) y + 1 = 0
(B) (p + q) (x + y)  2 = 0
(C) (2pq  1) (px + qy  1) = (p2 + q2  1) (qx + py  1)
(D) none
Q.15
From (3 , 4) chords are drawn to the circle x² + y²  4x = 0 . The locus of the mid points of the
chords is :
(A) x² + y²  5x  4y + 6 = 0
(B) x² + y² + 5x  4y + 6 = 0
(C) x² + y²  5x + 4y + 6 = 0
(D) x² + y²  5x  4y  6 = 0
Q.16
The lines y  y1 = m (x  x1) ± a 1  m2 are tangents to the same circle . The radius of the circle is :
(A) a/2
(B) a
(C) 2a
(D) none
Q.17
The centre of the smallest circle touching the circles x² + y²  2y  3 = 0 and
x² + y²  8x  18y + 93 = 0 is :
(A) (3 , 2)
(B) (4 , 4)
(C) (2 , 7)
(D) (2 , 5)
Q.18
If a, b, c are in harmonical progression then the line, bcx + cay + ab = 0 passes through a fixed point
whose coordinates are :
(A) (1, 2)
(B) ( 1, 2)
(C) ( 1,  2)
(D) (1,  2)
Q.19
A rhombus is inscribed in the region common to the two circles x 2 + y2  4x  12 = 0 and
x2 + y2 + 4x  12 = 0 with two of its vertices on the line joining the centres of the circles. The area of the
rhombous is :
(A) 8 3 sq.units
Q.20
(B) 4 3 sq.units
(C) 16 3 sq.units
(D) none
A variable straight line passes through the points of intersection of the lines, x + 2y = 1 and
Circle & Straight line
2x  y = 1 and meets the coordinate axes in A & B . The locus of the middle point of AB is :
(A) x + 3y  10xy = 0
(B) x  3y + 10xy = 0
(C) x + 3y + 10xy = 0
(D) none
Q.21
In a right triangle ABC, right angled at A, on the leg AC as diameter, a semicircle is described. The chord
joining A with the point of intersection D of the hypotenuse and the semicircle, then the length AC equals
to
(A)
AB  AD
AB 2  AD 2
(B)
AB  AD
AB  AD
(C)
AB  AD
(D)
AB  AD
AB2  AD2
Q.22
A variable straight line passes through a fixed point (a, b) intersecting the coordinates axes at A & B. If
'O' is the origin then the locus of the centroid of the triangle OAB is :
(A) bx + ay  3xy = 0
(B) bx + ay  2xy = 0
(C) ax + by  3xy = 0
(D) none
Q.23
The equation of the circle having the lines y2  2y + 4x  2xy = 0 as its normals & passing through the
point (2 , 1) is :
(A) x2 + y2  2x  4y + 3 = 0
(B) x2 + y2  2x + 4y  5 = 0
2
2
(C) x + y + 2x + 4y  13 = 0
(D) none
Q.24
If P = (1, 0) ; Q = (1, 0) & R = (2, 0) are three given points, then the locus of the points S satisfying
the relation, SQ2 + SR2 = 2 SP2 is :
(A) a straight line parallel to xaxis
(B) a circle passing through the origin
(C) a circle with the centre at the origin
(D) a straight line parallel to yaxis .
Q.25
If a circle passes through the point (a , b) & cuts the circle x² + y² = K² orthogonally, then the
equation of the locus of its centre is :
(A) 2ax + 2by  (a² + b² + K²) = 0
(B) 2ax + 2by  (a²  b² + K²) = 0
(C) x² + y²  3ax  4by + (a² + b²  K²) = 0 (D) x² + y²  2ax  3by + (a²  b²  K²) = 0
Q.26
The coordinates of the orthocentre of the triangle bounded by the lines, 4x  7y + 10 = 0; x + y=5 and
7x + 4y = 15 is :
(A) (2, 1)
(B) ( 1, 2)
(C) (1, 2)
(D) (1,  2)
Q.27
The distance between the chords of contact of tangents to the circle ; x2+ y2 + 2gx+2fy+ c=0 from
the origin & the point (g , f) is :
(A) g  f
2
2
(B)
g2  f 2  c
2
(C)
g2  f 2  c
2 g2  f 2
(D)
g2  f 2  c
2 g2  f 2
Q.28
The equation of the pair of bisectors of the angles between two straight lines is,
12x2  7xy  12y2 = 0 . If the equation of one line is 2y  x = 0 then the equation of the other line is :
(A) 41x  38y = 0
(B) 38x  41y = 0
(C) 38x + 41y = 0
(D) 41x + 38y = 0
Q.29
The points A (a , 0) , B (0 , b) , C (c , 0) & D (0 , d) are such that ac = bd & a, b, c, d are all non-zero.
Then the points :
(A) form a parallelogram
(B) do not lie on a circle
(C) form a trapezium
(D) are concyclic
Q.30
The line joining two points A (2, 0) ; B (3, 1) is rotated about A in the anticlock wise direction through
Circle & Straight line
an angle of 15º . The equation of the line in the new position is :
(A) x 
3 y  2 = 0 (B) x  2y  2 = 0
(C)
3xy2 3 =0
(D) none
Q.31
The locus of the centers of the circles which cut the circles x 2 + y2 + 4x  6y + 9 = 0 and
x2 + y2  5x + 4y  2 = 0 orthogonally is
(A) 9x + 10y  7 = 0
(B) x  y + 2 = 0
(C) 9x  10y + 11 = 0
(D) 9x + 10y + 7 = 0
Q.32
Area of the rhombus bounded by the four lines, ax ± by ± c = 0 is :
(A)
Q.33
Q.34
c2
2 ab
(B)
2 c2
ab
(C)
4 c2
ab
(D)
ab
4 c2
Given A  (1, 1) and AB is any line through it cutting the x-axis in B. If AC is perpendicular to AB and
meets the y-axis in C, then the equation of locus of mid- point P of BC is
(A) x + y = 1
(B) x + y = 2
(C) x + y = 2xy
(D) 2x + 2y = 1
The locus of the centers of the circles such that the point (2 , 3) is the mid point of the chord
5x + 2y = 16 is :
(A) 2x  5y + 11 = 0
(B) 2x + 5y  11 = 0
(C) 2x + 5y + 11 = 0
(D) none
Q.35
A stick of length 10 units rests against the floor & a wall of a room . If the stick begins to slide on the floor
then the locus of its middle point is :
(A) x2 + y2 = 2.5
(B) x2 + y2 = 25
(C) x2 + y2 = 100
(D) none
Q.36
The locus of the mid points of the chords of the circle x² + y² + 4x  6y  12 = 0 which subtend an angle
of

radians at its circumference is :
3
(A) (x  2)² + (y + 3)² = 6.25
(C) (x + 2)² + (y  3)² = 18.75
Q.37
(B) (x + 2)² + (y  3)² = 6.25
(D) (x + 2)² + (y + 3)² = 18.75
Through a given point P (a, b) a straight line is drawn to meet the axes at Q & R. If the parallelogram
OQSR is completed then the equation of the locus of S is
(given 'O' is the origin) :
(A)
a
b
+ =1
x
y
(B)
a
b
+ =1
y
x
(C)
a
b
+ =2
y
x
(D)
a
b
+ =2
y
x
Q.38
The points (x1, y1) , (x2, y2) , (x1, y2) & (x2, y1) are always :
(A) collinear
(B) concyclic
(C) vertices of a square
(D) vertices of a rhombus
Q.39
The number of possible straight lines , passing through (2, 3) and forming a triangle with coordinate axes,
whose area is 12 sq. units , is
(A) one
(B) two
(C) three
(D) four
Q.40
Two mutually perpendicular straight lines through the origin from an isosceles triangle with the line
2x + y = 5 . Then the area of the triangle is :
(A) 5
(B) 3
(C) 5/2
(D) 1
Q.41
The angle at which the circles (x – 1)2 + y2 = 10 and x2 + (y – 2)2 = 5 intersect is
Circle & Straight line
(A)

6
(B)

4
(C)

3
(D)

2
Q.42
A pair of straight lines x2 – 8x + 12 = 0 and y2 – 14y + 45 = 0 are forming a square.
Co-ordinates of the centre of the circle inscribed in the square are
(A) (3, 6)
(B) (4, 7)
(C) (4, 8)
(D) none
Q.43
The value of 'c' for which the set, {(x, y)x2 + y2 + 2x  1}  {(x, y)x  y + c  0} contains only
one point in common is :
(A) (, 1]  [3, )
(B) {1, 3}
(C) {3}
(D) { 1 }
Q.44
Co-ordinates of the orthocentre of the triangle whose vertices are A(0, 0) , B(3, 4) and C(4, 0) is
(A) (3, 1)
(B) (3, 4)
 3
(D)  3, 
 4
(C) (3, 3)
Q.45
Three lines x + 2y + 3 = 0 ; x + 2y – 7 = 0 and 2x – y – 4 = 0 form the three sides of two squares. The
equation to the fourth side of each square is
(A) 2x – y + 14 = 0 & 2x – y + 6 = 0
(B) 2x – y + 14 = 0 & 2x – y – 6 = 0
(C) 2x – y – 14 = 0 & 2x – y – 6 = 0
(D) 2x – y – 14 = 0 & 2x – y + 6 = 0
Q.46
P is a point (a, b) in the first quadrant. If the two circles which pass through P and touch both the
co-ordinate axes cut at right angles, then :
(A) a2  6ab + b2 = 0
(B) a2 + 2ab  b2 = 0
(C) a2  4ab + b2 = 0
(D) a2  8ab + b2 = 0
Q.47
If the vertices P and Q of a triangle PQR are given by (2, 5) and (4, –11) respectively, and the point R
moves along the line N: 9x + 7y + 4 = 0, then the locus of the centroid of the triangle PQR is a straight
line parallel to
(A) PQ
(B) QR
(C) RP
(D) N
Q.48
The range of values of 'a' such that the angle  between the pair of tangents drawn from the point
(a, 0) to the circle x2 + y2 = 1 satisfies
(A) (1, 2)

(B) 1 , 2


<  <  is :
2


(C)  2 ,  1

 
(D)  2 ,  1  1 , 2

Q.49
The points A(a, 0), B(0, b), C(c, 0) & D(0, d) are such that ac = bd & a, b, c, d are all nonzero. The
the points :
(A) form a parallelogram
(B) do not lie on a circle
(C) form a trapezium
(D) are concyclic
Q.50
If (,) is a point on the circle whose centre is on the x -axis and which touches the line x + y = 0 at
(2, –2), then the greatest value of  is
(A) 4 –
Q.51
2
(B) 6
(C) 4 + 2 2
(D) 4 +
2
Find the distance of the point (2, 5) from the line 3x + y + 4 = 0 measured parallel to the line 3x  4y +
Circle & Straight line
8=0.
[ans. 05 ]
Q.52
Three concentric circles of which the biggest is x 2 + y2 = 1, have their radii in A.P. If the line
y = x + 1 cuts all the circles in real and distinct points. The interval in which the common difference of the
A.P. will lie is



1
4
(B)  0 ,
(A)  0 , 
Q.53
Given

1 

2 2

(C)  0 ,

2  2

4 
(D) none
x y
 = 1 and ax + by = 1 are two variable lines, 'a' and 'b' being the parameters connected by
a b
the relation a2 + b2 = ab. The locus of the point of intersection has the equation
(A) x2 + y2 + xy  1 = 0
(B) x2 + y2 – xy + 1 = 0
(C) x2 + y2 + xy + 1 = 0
(D) x2 + y2 – xy – 1 = 0
Q.54
The chord of contact of the tangents drawn from a point on the circle, x 2 + y2 = a2 to the circle
x2 + y2 = b2 touches the circle x2 + y2 = c2 then a, b, c are in :
(A) A.P.
(B) G.P.
(C) H.P.
(D) A.G.P.
Q.55
A light beam emanating from the point A(3, 10) reflects from the line 2x + y - 6 = 0 and then passes
through the point B(5, 6) . The equation of the incident and reflected beams are respectively :
(A) 4 x  3 y + 18 = 0 & y = 6
(B) x  2 y + 8 = 0 & x = 5
(C) x + 2 y  8 = 0 & y = 6
(D) none of these
Q.56
If the two circles, x2 + y2 + 2 g1x + 2 f1y = 0 & x2 + y2 + 2 g2x + 2 f2y = 0 touch each then:
(A) f1 g1 = f2 g2
Q.57
(B)
 1

, p ; Q =
 xp 
If P  
k  N, then :
(A) Ar. ( PQR) =
f1
f
= 2
g1
g2
 1


, q ; R =
 xq 
p2 q2 r2
2
(C) f1 f2 = g1 g2
(D) none
 1 
 , r where xk  0, denotes the kth term of an H.P. for
 xr 
(p  q) 2  (q  r )2  ( r  p) 2
(B)  PQR is a right angled triangle
(C) the points P, Q, R are collinear
(D) none
Q.58
Tangents are drawn to the circle x2 + y2 = 1 at the points where it is met by the circles,
x2 + y2  ( + 6) x + (8  2 ) y  3 = 0 .  being the variable . The locus of the point of intersection of
these tangents is :
(A) 2x  y + 10 = 0
(B) x + 2y  10 = 0
(C) x  2y + 10 = 0
(D) 2x + y  10 = 0
Q.59
The acute angle between two straight lines passing through the point M( 6,  8) and the points in which
the line segment 2x + y + 10 = 0 enclosed between the co-ordinate axes is divided in the ratio
1 : 2 : 2 in the direction from the point of its intersection with the x  axis to the point of intersection with
the y  axis is :
(A) /3
(B) /4
(C) /6
(D) /12
Circle & Straight line
Q.60
B & C are fixed points having coordinates (3, 0) and ( 3, 0) respectively . If the vertical angle BAC
is 90º, then the locus of the centroid of the  ABC has the equation :
(A) x2 + y2 = 1
(B) x2 + y2 = 2
(C) 9 (x2 + y2) = 1
(D) 9 (x2 + y2) = 4
Q.61
Chords of the curve 4x2 + y2  x + 4y = 0 which subtend a right angle at the origin pass through a fixed
point whose co-ordinates are :
1
5
4
5
 1 4
 5 5
(A)  ,  
Q.62


(B)   , 
1 
a 
1
b


1
c


 1 4
 5 5
(C)  , 
 1
 5
4
5
(D)   ,  
1
d
If  a ,  ,  b ,  ,  c ,  &  d ,  are four distinct points on a circle of radius 4 units then,
abcd is equal to
(A) 4
(B) 1/4
(C) 1
(D) 16
Q.63
The pair of straight lines x2  4xy + y2 = 0 together with the line x + y + 4 6 = 0 form a triangle which
is :
(A) right angled but not isosceles
(B) right isosceles
(C) scalene
(D) equilateral
Q.64
If two chords, each bisected by the x  axis can be drawn to the circle,
2 (x2 + y2)  2ax  by = 0 (a  0 , b  0) from the point (a, b/2) then :
(A) a2 > 8b2
(B) b2 > 2a2
2
2
(C) a > 2b
(D) a2 = 2b2
Q.65
If the line y = mx bisects the angle between the lines ax 2 + 2h xy + by2 = 0 then m is a root of the
quadratic equation :
(A) hx2 + (a  b) x  h = 0
(B) x2 + h (a  b) x  1 = 0
2
(C) (a  b) x + hx  (a  b) = 0
(D) (a  b) x2  hx  (a  b) = 0
Q.66
Tangents are drawn to a unit circle with centre at the origin from each point on the line 2x + y = 4. Then
the equation to the locus of the middle point of the chord of contact is
(A) 2 (x2 + y2) = x + y
(B) 2 (x2 + y2) = x + 2 y
2
2
(C) 4 (x + y ) = 2x + y
(D) none
An equilateral triangle has each of its sides of length 6 cm . If (x1, y1) ; (x2, y2) & (x3, y3) are its vertices
then the value of the determinant,
Q.67
x1
x2
x3
(A) 192
y1 1
y2 1
y3 1
2
is equal to :
(B) 243
(C) 486
(D) 972
Q.68 Two circles whose radii are equal to 4 and 8 intersect at right angles. The length of their common chord is
(A)
Q.69
16
5
(B) 8
(C) 4 6
(D)
8 5
5
Points A & B are in the first quadrant ; point 'O' is the origin . If the slope of OA is 1, slope of OB is 7
and OA = OB, then the slope of AB is :
(A)  1/5
(B)  1/4
(C)  1/3
(D)  1/2
Circle & Straight line
Q.70
The common chord of two intersecting circles c1 & c2 can be seen from their centres at the angles of
90º and 60º respectively . If the distance between their centres is equal to
3 + 1 then the radii of c1 & c2 are :
(A)
3 &3
(B)
2 & 2 2
(C)
2 &2
(D) 2 2 & 4
Q.71
The co-ordinates of a point P on the line 2x  y + 5 = 0 such that PA  PB is maximum where A
is (4,  2) and B is (2,  4) will be :
(A) (11, 27)
(B) ( 11,  17)
(C) ( 11, 17)
(D) (0, 5)
Q.72
Three circles lie on a plane so that each of them externally touches the other two. Two of them has radius
3, the third having radius unity . If A, B & C are the points of tangency of the circles then the area of the
triangle ABC is
(A)
9 7
4
(B)
9 7
8
(C)
9 3
16
(D) none
Q.73
Let the co-ordinates of the two points A & B be (1, 2) and (7, 5) respectively. The line AB is rotated
through 45º in anti clockwise direction about the point of trisection of AB which is nearer to B. The
equation of the line in new position is :
(A) 2x  y  6 = 0
(B) x  y  1 = 0
(C) 3x  y  11 = 0
(D) none of these
Q.74
A pair of tangents are drawn to a unit circle with centre at the origin and these tangents intersect at A
enclosing an angle of 60°. The area enclosed by these tangents and the arc of the circle is
(A)
2

–
3
6
(B)

3 – 3
(C)

3
–
3
6


(D) 3 1  
6

Q.75
The true set of real values of  for which the point P with co-ordinate (, 2) does not lie inside the
triangle formed by the lines, x  y = 0 ; x + y  2 = 0 & x + 3 = 0 is :
(A) (,  2]
(B) [0, ]
(C) [ 2, 0]
(D) (,  2]  [0, ]
Q.76
If the line x cos  + y sin  = 2 is the equation of a transverse common tangent to the circles
Q.77
x2 + y2 = 4 and x2 + y2  6 3 x  6y + 20 = 0, then the value of  is :
(A) 5/6
(B) 2/3
(C) /3
(D) /6
2
The graph of the function, cos x cos (x + 2)  cos (x + 1) is :
(A) a straight line passing through (0 ,  sin2 1) with slope 2
(B) a straight line passing through (0 , 0)
(C) a parabola with vertex (1 ,  sin2 1)


,  sin 2 1 & parallel to the xaxis .
2

(D) a straight line passing through the point 
Q.78
A circle is drawn with y-axis as a tangent and its centre at the point which is the reflection of (3, 4) in the
line y = x. The equation of the circle is
(A) x2 + y2 – 6x – 8y + 16 = 0
(B) x2 + y2 – 8x – 6y + 16 = 0
(C) x2 + y2 – 8x – 6y + 9 = 0
(D) x2 + y2 – 6x – 8y + 9 = 0
Circle & Straight line
Q.79
Let PQR be a right angled isosceles triangle, right angled at P (2, 1). If the equation of the line QR is
2x + y = 3, then the equation representing the pair of lines PQ and PR is
(A) 3x2  3y2 + 8xy + 20x + 10y + 25 = 0
(B) 3x2  3y2 + 8xy  20x  10y + 25 = 0
(C) 3x2  3y2 + 8xy + 10x + 15y + 20 = 0
(D) 3x2  3y2  8xy  10x  15y  20 = 0
Q.80
A circle of constant radius ' a ' passes through origin ' O ' and cuts the axes of coordinates in points P
and Q, then the equation of the locus of the foot of perpendicular from O to PQ is :
Q.81
(A) (x2 + y2)  12  12  = 4 a2
x
y 
(B) (x2 + y2)2  12  12  = a2
x
y 
(C) (x2 + y2)2  12  12  = 4 a2
x
y 
(D) (x2 + y2)  12  12  = a2
x
y 
4
A is a point on either of two lines y + 3 x= 2 at a distance of
units from their point of intersection.
3
The co-ordinates of the foot of perpendicular from A on the bisector of the angle between them are


(A)  

, 2
3 
2
 2

, 2
 3 
(B) (0, 0)
(C) 
(D) (0, 4)
Q.82
The circle passing through the distinct points (1, t) , (t, 1) & (t, t) for all values of ' t ' , passes through
the point :
(A) ( 1,  1)
(B) ( 1, 1)
(C) (1,  1)
(D) (1, 1)
Q.83
In a triangle ABC, side AB has the equation 2 x + 3 y = 29 and the side AC has the equation ,
x + 2 y = 16 . If the mid  point of BC is (5, 6) then the equation of BC is :
(A) x  y =  1
(B) 5 x  2 y = 13
(C) x + y = 11
(D) 3 x  4 y =  9
Q.84
If a circle of constant radius 3k passes through the origin 'O' and meets co-ordinate axes at A and B
then the locus of the centroid of the triangle OAB is
(A) x2 + y2 = (2k)2
(B) x2 + y2 = (3k)2
(C) x2 + y2 = (4k)2
(D) x2 + y2 = (6k)2
Q.85
The circumcentre of the triangle formed by the lines , x y + 2 x + 2 y + 4 = 0 and x + y + 2 = 0 is
(A) ( 2,  2)
(B) ( 1,  1)
(C) (0, 0)
(D) ( 1,  2)
Q.86
The locus of the midpoints of the chords of the circle x2 + y2  2x  4y  11 = 0 which subtend 600 at
the centre is
(A) x2 + y2  4x  2y  7 = 0
(B) x2 + y2 + 4x + 2y  7 = 0
2
2
(C) x + y  2x  4y  7 = 0
(D) x2 + y2 + 2x + 4y + 7 = 0
Q.87
ABC is an isosceles triangle . If the co-ordinates of the base are (1, 3) and ( 2, 7) ,
then co-ordinates of vertex A can be :


(A)  1 , 5
2
Q.88


(B)  1 , 5
8
(C)
 65 ,  5

(D)  7 , 1
8

Tangents are drawn from (4, 4) to the circle x2 + y2  2x  2y  7 = 0 to meet the circle at A and B. The
length of the chord AB is
(A) 2 3
(B) 3 2
(C) 2 6
Circle & Straight line
(D) 6 2
Q.89
The line x + y = p meets the axis of x & y at A & B respectively . A triangle APQ is inscribed in the
triangle OAB, O being the origin, with right angle at Q . P and Q lie respectively on OB and AB . If the
area of the triangle APQ is 3/8th of the area of the triangle OAB, then
AQ
is equal to :
BQ
(A) 2
(D) 3
(B) 2/3
(C) 1/3
Q.90
The equation of the image of the circle x 2 + y2 + 16x  24y + 183 = 0 by the line mirror
4x + 7y + 13 = 0 is:
(A) x2 + y2 + 32x  4y + 235 = 0
(B) x2 + y2 + 32x + 4y  235 = 0
(C) x2 + y2 + 32x  4y  235 = 0
(D) x2 + y2 + 32x + 4y + 235 = 0
Q.91
If in triangle ABC , A  (1, 10) , circumcentre   13 , 23
co-ordinates of mid-point of side opposite to A is :
(A) (1,  11/3)
(B) (1, 5)
(C) (1,  3)




,4
and orthocentre  11
3 3 then the
(D) (1, 6)
Q.92
Let x & y be the real numbers satisfying the equation x2  4x + y2 + 3 = 0. If the maximum and minimum
values of x2 + y2 are M & m respectively, then the numerical value of M  m is :
(A) 2
(B) 8
(C) 15
(D) none of these
Q.93
If the straight lines , ax + amy + 1 = 0 , b x + (m + 1) b y + 1 = 0 and cx + (m + 2)cy + 1 = 0,
m  0 are concurrent then a, b, c are in :
(A) A.P. only for m = 1
(B) A.P. for all m
(C) G.P. for all m
(D) H.P. for all m.
Q.94
A line meets the co-ordinate axes in A & B. A circle is circumscribed about the triangle OAB. If d1 & d2
are the distances of the tangent to the circle at the origin O from the points A and B respectively, the
diameter of the circle is :
(A)
2d1  d 2
2
(B)
d1  2d 2
2
(C) d1 + d2
d1d 2
(D) d  d
1
2
Q.95
If x1 , y1 are the roots of x2 + 8 x  20 = 0 , x2 , y2 are the roots of 4 x2 + 32 x  57 = 0 and
x3 , y3 are the roots of 9 x2 + 72 x  112 = 0 , then the points, (x1 , y1) , (x2 , y2) & (x3 , y3)
(A) are collinear
(B) form an equilateral triangle
(C) form a right angled isosceles triangle
(D) are concyclic
Q.96
Two concentric circles are such that the smaller divides the larger into two regions of equal area. If the
radius of the smaller circle is 2 , then the length of the tangent from any point ' P ' on the larger circle to
the smaller circle is :
(A) 1
Q.97
(B) 2
(C) 2
(D) none
Triangle formed by the lines x + y = 0 , x – y = 0 and lx + my = 1. If l and m vary subject to the
condition l 2 + m2 = 1 then the locus of its circumcentre is
(A) (x2 – y2)2 = x2 + y2
(B) (x2 + y2)2 = (x2 – y2)
(C) (x2 + y2) = 4x2 y2
(D) (x2 – y2)2 = (x2 + y2)2
Circle & Straight line
Q.98
The equation of a line inclined at an angle

to the axis X, such that the two circles
4
x2 + y2 = 4, x2 + y2 – 10x – 14y + 65 = 0 intercept equal lengths on it, is
(A) 2x – 2y – 3 = 0
(B) 2x – 2y + 3 = 0
(C) x – y + 6 = 0
(D) x – y – 6 = 0
Q.99
The coordinates of three points A(4, 0) ; B(2, 1) and C(3, 1) determine the vertices of an equilateral
trapezium ABCD . The coordinates of the vertex D are :
(A) (6, 0)
(B) ( 3, 0)
(C) ( 5, 0)
(D) (9, 0)
Q.100 Tangents are drawn from any point on the circle x2 + y2 = R2 to the circle x2 + y2 = r2. If the line joining
the points of intersection of these tangents with the first circle also touch the second, then R equals
(A) 2 r
(B) 2r
(C)
2r
2 3
(D)
4r
3 5
Q.101 The image of the pair of lines represented by ax2 + 2h xy + by2 = 0 by the line mirror y = 0 is
(A) ax2  2h xy  by2 = 0
(B) bx2  2h xy + ay2 = 0
(C) bx2 + 2h xy + ay2 = 0
(D) ax2  2h xy + by2 = 0
Q.102 Pair of tangents are drawn from every point on the line 3x + 4y = 12 on the circle x 2 + y2 = 4. Their
variable chord of contact always passes through a fixed point whose co-ordinates are
4 3
(A)  , 
3 4
3 3
(B)  , 
4 4
(C) (1, 1)
 4
(D) 1, 
 3
Q.103 The set of values of 'b' for which the origin and the point (1, 1) lie on the same side of the straight line,
a2x + a by + 1 = 0  a  R, b > 0 are :
(A) b  (2, 4)
(B) b  (0, 2)
(C) b  [0, 2]
(D) (2, )
Q.104 The equation of the circle symmetric to the circle x 2 + y2 – 2x – 4y + 4 = 0 about the line
x – y = 3 is
(A) x2 + y2 – 10x + 4y + 28 = 0
(B) x2 + y2 + 6x + 8 = 0
(C) x2 + y2 – 14x – 2y + 49 = 0
(D) x2 + y2 + 8x + 2y + 16 = 0
Q.105 Which one of the following statement is True ?
(A) The lines 2x + 3y + 19 = 0 and 9x + 6y  17 = 0 cut the coordinate axes in concyclic points.
(B) The circumcentre, orthocentre, incentre and centroid of the triangle formed by the points A(1, 2) ,
B(4, 6) , C( 2,  1) are colinear .
(C) The mid point of the sides of a triangle are (1, 2) , (3, 1) & (5, 5) . The orthocentre of the triangle
has the coordinates (3, 1) .
(D) Equation of the line pair through the origin and perpendicular to the line pair
x y  3 y2 + y  2 x + 10 = 0 is 3 y2 + x y = 0
Q.106 The locus of the centre of a circle which touches externally the circle ,
x² + y²  6x  6y + 14 = 0 & also touches the y-axis is given by the equation :
(A) x²  6x  10y + 14 = 0
(B) x²  10x  6y + 14 = 0
(C) y²  6x  10y + 14 = 0
(D) y²  10x  6y + 14 = 0
Circle & Straight line
Q.107 The co-ordinates of the vertices P, Q, R & S of square PQRS inscribed in the triangle ABC with vertices
A  (0, 0) , B  (3, 0) & C  (2, 1) given that two of its vertices P, Q are on the side AB are respectively
1
4
 3
 8
  3 1
  8 8
 1 1
 4 8
1
3
3 1
1 1
(B)  , 0 ,  , 0 ,  ,  &  , 
1
2
3
9
9 3
3 3
(D)  , 0 ,  , 0 ,  ,  &  , 
(A)  , 0 ,  , 0 ,  ,  &  , 
3
2
  3 1
  3 2


(C) (1, 0) ,  , 0 ,  ,  &  1 , 
2
2
 4
 4
  4 4
  4 4
 2 4
 2 4
Q.108 The equation of the locus of the mid points of the chords of the circle
2
4x2 + 4y2  12x + 4y + 1 = 0 that subtend an angle of
at its centre is
3
(A) 16(x² + y²)  48x + 16y + 31 = 0
(B) 16(x² + y²)  48x – 16y + 31 = 0
(C) 16(x² + y²) + 48x + 16y + 31 = 0
(D) 16(x² + y²) + 48x – 16y + 31 = 0
Q.109 The line 2x + 3y = 12 meets the x - axis at A and the y - axis at B . The line through (5, 5) perpendicular
to AB meets the x - axis, y - axis & the line AB at C, D, E respectively. If O is the origin, then the area of
the OCEB is :
(A)
20
sq. units
3
(B)
23
sq. units
3
(C)
26
sq. units
3
(D)
5 52
sq. units
9
Q.110 In the xy plane, the segment with end points (3, 8) and (–5, 2) is the diameter of the circle. The point
(k, 10) lies on the circle for
(A) no value of k
(B) exactly one integral k
(C) exacly one non integral k
(D) two real values of k
Q.111 Let A  (3, 2) and B  (5, 1). ABP is an equilateral triangle is constructed on the side of AB remote from
the origin then the orthocentre of triangle ABP is
1
3


3,  3
(A)  4 


2
2
1
3


3,  3 
(B)  4 


2
2
1
3 1 

3, 
3
(C)  4 

6
2 3 
1
3 1 

3, 
3
(D)  4 

6
2 3 
Q.112 The vertex of a right angle of a right angled triangle lies on the straight line 2x + y – 10 = 0 and the two
other vertices, at points (2, –3) and (4, 1) then the area of triangle in sq. units is
(A) 10
(B) 3
(C)
33
5
Circle & Straight line
(D) 11
Select the correct alternatives : (More than one are correct)
Q.113
Let u  ax + by + a 3 b = 0 v  bx  ay + b 3 a = 0 a, b  R be two straight lines. The equation
of the bisectors of the angle formed by k1u  k2v = 0 & k1u + k2v = 0 for non zero real k1 & k2 are:
(A) u = 0
(B) k2u + k1v = 0
(C) k2u  k1v = 0
(D) v = 0
Q.114 A tangent drawn from the point (4, 0) to the circle x2 + y2 = 8 touches it at a point A in the first quadrant.
The coordinates of another point B on the circle such that l (AB) = 4 are :
(A) (2,  2)

(C)  2 2 , 0
(B) ( 2, 2)


(D) 0 ,  2 2

Q.115 Consider the equation y  y1 = m (x  x1) . If m & x1 are fixed and different lines are drawn for different
values of y1, then :
(A) the lines will pass through a fixed point (B) there will be a set of parallel lines
(C) all the lines intersect the line x = x1
(D) all the lines will be parallel to the line y = x1.
Q.116 A circle passes through the points ( 1, 1) , (0, 6) and (5, 5) . The point(s) on this circle, the tangent(s)
at which is/are parallel to the straight line joining the origin to its centre is/are :
(A) (1,  5)
(B) (5, 1)
(C) ( 5,  1)
(D) ( 1, 5)
Q.117 If one vertex of an equilateral triangle of side 'a' lies at the origin and the other lies on the line
x  3 y = 0 then the co-ordinates of the third vertex are :
 3a
a


,

(B)  2
2 

(A) (0, a)

3a a


(D)  2 , 2 


(C) (0,  a)
Q.118 Equation of a line through (7, 4) and touching the circle, x2 + y2  6x + 4y  3 = 0 is :
(A) 5x  12y + 13 = 0
(B) 12x  5y  64 = 0
(C) x  7 = 0
(D) y = 4
Q.119 Three vertices of a triangle are A(4, 3) ; B(1,  1) and C(7, k) . Value(s) of k for which centroid,
orthocentre, incentre and circumcentre of the  ABC lie on the same straight line is/are :
(A) 7
(B)  1
(C)  19/8
(D) none
Q.120 Point M moved along the circle (x  4)2 + (y  8)2 = 20 . Then it broke away from it and moving along
a tangent to the circle, cuts the xaxis at the point ( 2, 0) . The coordinates of the point on the circle
at which the moving point broke away can be :
(A)   3 , 46 
 5
5
2 44
(B)   , 
 5
5
(C) (6, 4)
(D) (3, 5)
Q.121 Straight lines 2x + y = 5 and x  2y = 3 intersect at the point A . Points B and C are chosen on these
two lines such that AB = AC . Then the equation of a line BC passing through the point (2, 3) is
(A) 3x  y  3 = 0
(B) x + 3y  11 = 0
(C) 3x + y  9 = 0
(D) x  3y + 7 = 0
Q.122 The centre(s) of the circle(s) passing through the points (0, 0) , (1, 0) and touching the circle
x2 + y2 = 9 is/are :
 3 1
(A)  , 
2 2
 1 3
(B)  , 
2 2
1

1

1/ 2
1/ 2
(C)  , 2  (D)  ,  2 
2
2
Circle & Straight line
Q.123 The x  co-ordinates of the vertices of a square of unit area are the roots of the equation
x2  3x + 2 = 0 and the y  co-ordinates of the vertices are the roots of the equation
y2  3y + 2 = 0 then the possible vertices of the square is/are :
(A) (1, 1), (2, 1), (2, 2), (1, 2)
(B) ( 1, 1), ( 2, 1), ( 2, 2), ( 1, 2)
(C) (2, 1), (1,  1), (1, 2), (2, 2)
(D) ( 2, 1), ( 1,  1), ( 1, 2), ( 2, 2)

Q.124 A circle passes through the point  3 ,

7
 and touches the line pair x2  y2  2x + 1 = 0. The
2 
co-ordinates of the centre of the circle are :
(A) (4, 0)
(B) (5, 0)
(C) (6, 0)
(D) (0, 4)
Q.125 P (x, y) moves such that the area of the triangle formed by P, Q (a , 2 a) and R ( a,  2 a) is equal to
the area of the triangle formed by P, S (a, 2 a) & T (2 a, 3 a). The locus of 'P' is a straight line given
by :
(A) 3x  y = a
(B) 5x  3y + a = 0
(C) y = 2ax
(D) 2y = ax
Circle & Straight line
Q.1
D
Q.2
C
Q.6
B
Q.7
Q.11
A
Q.16
ANSWER KEY
Q.3
A
Q.4
B
Q.5
A
C
Q.8
C
Q.9
B
Q.10
A
Q.12
C
Q.13
C
Q.14
C
Q.15
A
B
Q.17
D
Q.18
D
Q.19
A
Q.20
A
Q.21
D
Q.22
A
Q.23
A
Q.24
D
Q.25
A
Q.26
C
Q.27
C
Q.28
A
Q.29
D
Q.30
C
Q.31
C
Q.32
B
Q.33
A
Q.34
A
Q.35
B
Q.36
B
Q.37
A
Q.38
B
Q.39
C
Q.40
A
Q.41
B
Q.42
B
Q.43
D
Q.44
D
Q.45
D
Q.46
C
Q.47
D
Q.48
D
Q.49
D
Q.50
C
Q.51
C
Q.52
C
Q.53
A
Q.54
B
Q.55
A
Q.56
B
Q.57
C
Q.58
A
Q.59
B
Q.60
A
Q.61
A
Q.62
C
Q.63
D
Q.64
C
Q.65
A
Q.66
C
Q.67
D
Q.68
A
Q.69
D
Q.70
C
Q.71
B
Q.72
C
Q.73
C
Q.74
B
Q.75
D
Q.76
D
Q.77
D
Q.78
C
Q.79
B
Q.80
C
Q.81
B
Q.82
D
Q.83 C
Q.84
A
Q.85
B
Q.86
C
Q.87
D
Q.88
B
Q.89
D
Q.90
D
Q.91
A
Q.92
B
Q.93
D
Q.94
C
Q.95
A
Q.96
C
Q.97
A
Q.98
A
Q.99
D
Q.100 B
Q.101 D
Q.102 D
Q.103 B
Q.104 A
Q.105 A
Q.106 D
Q.107 D
Q.108 A
Q.109 B
Q.110 B
Q.111 D
Q.112 B
Q.113 A, D
Q.114 A, B
Q.115 B, C
Q.116 B, D
Q.117 A, B, C, D
Q.118 A, C
Q.119 B, C
Q.120 B, C
Q.121 A, B
Q.122 C, D
Q.123 A, B
Q.124 A, C
Q.125 A, B
Circle & Straight line