RACE # 80 CIRCLE

SINGLE CORRECT TYPE

1. The equation of the circle passing through (3, 6) and whose centre is (2, –1) is -
(A) x2 + y2 – 4x + 2y = 45 (B) x2 + y2 – 4x – 2y + 45 = 0
(C) x2 + y2 + 4x – 2y = 45 (D) x2 + y2 – 4x + 2y + 45 = 0
2. The radius of the circle passing through the points (0, 0), (1, 0) and (0, 1) is-
1 1
(A) 2 (B) (C) 2 (D)
2 2
3. If a be the radius of a circle which touches x-axis at the origin, then its equation is
(A) x2 + y2 + ax = 0 (B) x2 + y2 ± 2ya = 0
(C) x2 + y2 ± 2xa = 0 (D) x2 + y2 + ya = 0
4. The equation of a circle which passes through the three points (3, 0) (1, –6), (4, –1) is
(A) 2x2 + 2y2 + 5x – 11y + 3 = 0 (B) x2 + y2 – 5x + 11y – 3 = 0
(C) x2 + y2 + 5x – 11y + 3 = 0 (D) 2x2 + 2y2 – 5x + 11y – 3 = 0
5. If y = 3 x + c1 & y = 3 x + c2 are two parallel tangents of a circle of radius 2 units, then |c1 – c2|
is equal to
(A) 8 (B) 4 (C) 2 (D) 1
6. Equation 2x + 2y + 2 lx + l = 0 represents a circle for :
2 2 2

(A) each real value of l (B) no real value of l

(C) positive l (D) negative l
7. A circle of radius 5 has its centre on the negative x-axis and passes through the point (2, 3).
The intercept made by the circle on the y-axis is

(A) 10 (B) 2 21 (C) 2 11 (D) imaginary y-intercept

8. A straight line l1 with equation x – 2y + 10 = 0 meets the circle with equation x2 + y2 = 100 at
B in the first quadrant. A line through B, perpendicular to l1 cuts the y-axis at P (0, t). The
value of 't' is
(A) 12 (B) 15 (C) 20 (D) 25
SUBJECTIVE TYPE QUESTIONS
9. ABCD in order be a rectangle. The co-ordinate of A and C are (1, 3) and (5, 1) respectively.
If the gradient of the diagonal BD is 2 and the area of the rectangle is S, then the value
S
of is equal to
10
10. From point (–1, 4), perpendicular lines are drawn to form isosceles triangle with line y =
D
2x – 4 such that area of triangle formed is D, then is equal to
4
11. Find the coordinates of the centre and the radius of the circles whose equations are
(a) 3x2 + 3y2 – 5x – 6y + 4 = 0
(b) 4x2 + 4y2 – 16x – 12y + 21 = 0.
12. (a) If (4, 1) is an extremity of a diameter of the circle x2 + y2 – 2x + 6y – 15 = 0, find the
co-ordinates of the other extremity of the diameter.
(b) Show that the point (7, – 5) lies on the circle x + y – 6x + 4y – 12 = 0 and find the co-
2 2

ordinates of the other end of the diameter through this point.

1
RACE # 81 CIRCLE

SINGLE CORRECT TYPE

1. Consider the lines 2x – y – 3 = 0 and x – 3y + 5 = 0. Which of the following statement is
correct ?
(A) origin lies on bisector of acute angle
(B) origin lies in acute angle
(C) origin lies on bisector of obtuse angle
(D) origin lies in obtuse angle
2. If (x, 3) and (3, 5) are the extremities of a diameter of a circle with centre at (2, y). Then
the value of x and y are-
(A) x = 1, y = 4 (B) x = 4, y = 1
(C) x = 8, y = 2 (D) None of these
3. Four distinct points (2k, 3k), (1, 0), (0, 1) and (0,0) lie on a circle for -
(A) All integral values of k (B) 0 < k < 1
(C) k < 0 (D) k = 5/13
4. Equation of circle touching the lines |x| + |y| = 4 is -
(A) x2 + y2 = 12 (B) x2 + y2 = 16 (C) x2 + y2 = 4 (D) x2 + y2 = 8
5. The length of intercept on y-axis, by a circle whose diameter is the line joining the points
(–4,3) and (12,–1) is

(A) 3 2 (B) 13 (C) 4 13 (D) none of these

MULTIPLE CORRECT CHOICE TYPE
6. If the equation x2 + y2 + 2lx + 4 = 0 and x2 + y2 – 4ly + 8 = 0 represent real circles then the
value of l can be
(A) 1 (B) 2 (C) 3 (D) 5
SUBJECTIVE TYPE
7. Find the equation to the circle which touches each positive axes at a distance 5 from the
origin.
8. Find the equation to the circle which touches the axis of x and passes through the two points
(1, – 2) and (3, – 4).
9. Find the equation to the circle which circumscribes the triangle formed by the lines x = 3 ;
x + y + 3 = 0 and x – y + 1 = 0.
10. In DABC, AC > AB, the internal angle bisector of angle A meets BC at D and E (E is inside
the triangle) is the foot of the perpendicular from B on AD. Suppose AB = 5 and BE = 4,

æ AC + AB ö
the value of the expression ç ÷ (ED) is
è AC - AB ø

2
 MATRIX MATCH TYPE
11. Consider the expression ƒ(x) = (2sin q - 3)x2 + 3x - 1 .
Match the columns for the values of q given in column-II satisfying the conditions given
in column-I
Column-I Column-II
p
(A) The parabola represented by y = ƒ (x) is (P) q = np + (-1)n , n Î I
6
opening upwards
æ ö p
(B) One of the root of ƒ(x) = 0 is one (Q) q Î ç 2np, 2np + ÷
è 6 ø
æ p ö p
(C) One of root is greater than one & other root is smaller (R) q Î ç 2np + , 2np + ÷
è 6 3 ø
than one
æ p 2p ö
(D) Roots are real and of opposite sign (S) q Î ç 2np + , 2np +
è 3 3 ÷ø
æ 2p 5p ö
(T) q Î ç 2np + 3 , 2np + 6 ÷
è ø

3
RACE # 82 CIRCLE

SINGLE CORRECT TYPE

x y
1. The equation of the circum-circle of the triangle formed by the lines x = 0, y = 0, – = 1,
a b
is -
(A) x2 + y2 + ax – by = 0 (B) x2 + y2 – ax + by = 0
(C) x2 + y2 – ax – by = 0 (D) x2 + y2 + ax + by = 0
2. The circum-circle of the quadrilateral formed by the lines x = a, x = 2a, y = - a, y = a is-
(A) x2 + y2 – 3ax – a2 = 0 (B) x2 + y2 + 3ax + a2 = 0
(C) x2 + y2 – 3ax + a2 = 0 (D) x2 + y2 + 3ax – a2 = 0
3. The x coordinates of two points A and B are roots of equation x2 + 2x – a2 = 0 and y
coordinate are roots of equation y2 + 4y – b2 = 0 then equation of the circle which has
diameter AB is-

(A) (x – 1)2 + (y – 2)2 = 5 + a2 + b2 (B) (x + 1)2 + (y + 2)2 = (5 + a 2 + b2 )

(C) (x + 1)2 + (y + 2)2 = (a2 + b2) (D) (x + 1)2 + (y + 2)2 = 5 + a2 + b2

4. The equation of the circle touching the lines x = 0, y = 0 and x = 2c is-
(A) x2 + y2 + 2cx + 2cy + c2 = 0 (B) x2 + y2 – 2cx + 2cy + c2 = 0
(C) x2 + y2 ± 2cx – 2cy + c2 = 0 (D) x2 + y2 – 2cx ± 2cy + c2 = 0
5. The circle x2 + y2 – 4x – 4y + 4 = 0
(A) touches x-axis only (B) touches both axes
(C) passes through the origin (D) touches y-axis only
6. The equation of the in-circle of the triangle formed by the axes and the line 4x + 3y = 6
is -
(A) x2 + y2 – 6x – 6y + 9 = 0 (B) 4(x2 + y2 – x – y) + 1 = 0
(C) 4(x2 + y2 + x + y) + 1 = 0 (D) x2 + y2 – 6x – 6y – 9 = 0
7. The equation of the circle which passes through (1, 0) and (0, 1) and has its radius as small
as possible, is -
(A) 2x2 + 2y2 – 3x – 3y + 1 = 0 (B) x2 + y2 – x – y = 0
(C) x2 + y2 – 2x – 2y + 1 = 0 (D) x2 + y2 – 3x – 3y + 2 = 0
8. A circle is inscribed in an equilateral triangle of side 6. Find the area of any square inscribed
in the circle -
(A) 36 (B) 12 (C) 6 (D) 9

4
 MATRIX MATCH TYPE
9. Match List-I with List-II and select the correct answer using the code given below the
list.
List-I List-II
(P) Distance between the two lines (1) 1

x + 3y - 2 = 0 and x + 3y + 2 = 0 is
(Q) The variable line x + ly + l – 5 = 0 (l Î R) passes (2) 2
through a fixed point. The x co-ordinate of fixed point is
(R) The x-coordinates of the point of intersection of lines (3) 5
3x + 4y = 9 and y = mx + 1 is an integer, if m is also
an integer then sum of possible values of m is
(S) The x-coordinate of incentre of triangle, whose vertices (4) –3

( )
are 1, 3 , ( 0, 0 ) and (2,0), is

Codes :
P Q R S
(A) 2 3 1 4
(B) 2 3 4 1
(C) 3 2 4 1
(D) 1 2 4 3

5
RACE # 83 CIRCLE

1. The straight line (x – 2) + (y + 3) = 0 meets the circle (x – 2) 2 + (y – 3)2 = 11 at-

(A) zero points (B) two points (C) one point (D) None of these
2. If the line 3x + 4y = m touches the circle x2 + y2 = 10x, then m is equal to-
(A) 40, 10 (B) 40, –10 (C) –40, 10 (D) –40, –10
3. The length of the intercept made by the circle x2 + y2 =1 on the line x + y = 1 is-

(A) 1/ 2 (B) 2 (C) 2 (D) 2 2

4. If a circle with centre (0, 0) touches the line 5x + 12y = 1 then its equation will be-
(A) 13(x2 + y2) = 1 (B) x2 + y2 = 169 (C) 169(x2 + y2) = 1 (D) x2 + y2 = 13

x y
5. The equation of circle which touches the axes of coordinates and the line + = 1 and
3 4
whose centre lies in the first quadrant is x2 + y2 – 2cx – 2cy + c2 = 0, where c is-
(A) 2 (B) 0 (C) 3 (D) 6
6. The lines 12x – 5y – 17 = 0 and 24x – 10y + 44 =0 are tangents to the same circle. Then
the radius of the circle is-
1
(A) 1 (B) 1 (C) 2 (D) None of these
2
7. If the circle x2 + y2 = a2 cuts off a chord of length 2b from the line y = mx + c, then-
(A) (1–m2) (a2 – b2) = c2 (B) (1 + m2) (a2 – b2) = c2
(C) (1–m2) (a2 + b2) = c2 (D) (1 + m2) (a2 + b2) = c2
8. Locus of centre of circle touching the straight lines 3x + 4y = 5 and 3x + 4y = 20 is -
(A) 3x + 4y = 15 (B) 6x + 8y = 15 (C) 3x + 4y = 25 (D) 6x + 8y = 25
9. If the equation of the in-circle of an equilateral triangle is x + y2 + 4x – 6y + 4 = 0,
2

then equation of circum-circle of the triangle is-

(A) x2 + y2 + 4x + 6y – 23 = 0 (B) x2 + y2 + 4x – 6y – 23 = 0
(C) x2 + y2 – 4x – 6y – 23 = 0 (D) x2 + y2 + 4x – 6y + 23 = 0
p
10. The equation of a line inclined at an angle with positive x-axis, such that the two
4
circles 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

1 1 1
11. If æç a , ö÷ , æç b , ö÷ , æç c , 1 ö÷ and æç d , ö÷ are four distinct points on a circle of radius 4 units then,
è aø è bø è cø è dø

abcd is equal to
(A) 4 (B) 1/4 (C) 1 (D) 16
12. The ends of a quadrant of a circle have the coordinates (1, 3) and (3, 1) then the centre of the
such a circle is
(A) (1, 1) (B) (2, 2) (C) (2, 6) (D) (4, 4)

6
RACE # 84 CIRCLE

SINGLE CORRECT TYPE

1. Four unit circles pass through the origin and have their centres on the coordinate axes. The
area of the quadrilateral whose vertices are the points of intersection (in pairs) of the circles, is
(A) 1 sq. unit

(B) 2 2 sq. units

(C) 4 sq. units
(D) can not be uniquely determined, insufficient data
2. Consider 3 non collinear points A, B, C with coordinates (0, 6), (5, 5) and (–1, 1) respectively.
Equation of a line tangent to the circle circumscribing the triangle ABC and passing through
the origin is
(A) 2x – 3y = 0 (B) 3x + 2y = 0 (C) 3x – 2y = 0 (D) 2x + 3y = 0
3. From the point A (0 , 3) on the circle x² + 4x + (y - 3)² = 0 a chord AB is drawn and
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
4. The gradient of the tangent line at the point (a cos a, a sin a) to the circle x2 + y2 = a2, is
(A) tan(p–a) (B) tan a (C) cot a (D) – cot a
5. If y = c is a tangent to the circle x 2 + y 2 – 2x + 2y – 2 = 0 at (1, 1), then the value of c is-
(A) 1 (B) 2 (C) –1 (D) –2
6. Line 3x + 4y = 25 touches the circle x2 + y2 = 25 at the point
(A) (4, 3) (B) (3, 4) (C) (–3, –4) (D) (–4, –3)
7. The equations of the tangents drawn from the point (0, 1) to the circle x 2 + y2 – 2x + 4y = 0 are-
(A) 2x – y + 1 = 0, x + 2y – 2 = 0 (B) 2x – y – 1 = 0, x + 2y – 2 = 0
(C) 2x – y + 1 = 0, x + 2y + 2 = 0 (D) 2x – y – 1 = 0, x + 2y + 2 = 0
8. The equations of tangents to the circle x2 + y2 – 22x – 4y + 25 = 0 which are perpendicular to
the line 5x + 12y + 8 = 0 are
(A) 12x – 5y + 8 = 0, 12x – 5y = 252 (B) 12x – 5y – 8 = 0, 12x – 5y + 252 = 0
(C) 12x – 5y = 0, 12x – 5y = 252 (D) None of these
9. The equation of the normal to the circle x2 + y2 – 8x – 2y + 12 = 0 at the points whose ordinate
is –1, will be-
(A) 2x – y – 7 = 0, 2x + y – 9 = 0 (B) 2x + y – 7 = 0, 2x + y + 9 = 0
(C) 2x + y + 7 = 0, 2x + y + 9 = 0 (D) 2x – y + 7 = 0, 2x – y + 9 = 0
10. If the lengths of the tangents drawn from the point (1, 2) to the circles x2 + y2 + x + y – 4 = 0
and 3x2 + 3y2 – x – y + k =0 be in the ratio 4 : 3, then k is
(A) 21/2 (B) 7/2 (C)–21/4 (D) 7/4
11. The angle between the tangents drawn from the origin to the circle (x – 7)2 + (y + 1)2 = 25 is-
(A) p / 3 (B) p / 6 (C) p / 2 (D) p / 8

7
RACE # 85 CIRCLE

SINGLE CORRECT TYPE

1. The equation of the chord of contact, if the tangents are drawn from the point (5, –3) to the
circle x2 + y2 = 10, is-
(A) 5x – 3y = 10 (B) 3x + 5y = 10 (C) 5x + 3y = 10 (D) 3x – 5y = 10
2. Two perpendicular tangents to the circle x2 + y2 = a2 meet at P. Then the locus of P has the
equation-
(A) x2 + y2 = 2a2 (B) x2 + y2 = 3a2 (C) x2 = y2 = 4a2 (D) None of these
3. Find the locus of mid point of chords of circle x2 + y2 = 25 which subtends right angle at origin-
(A) x2 + y2 = 25/4 (B) x2 + y2 = 5 (C) x2 + y2 = 25/2 (D) x2 + y2 = 5/2
4. The equation of the chord of the circle x2 + y2 – 6x + 8y = 0 which is bisected at the point (5, –3)
is-
(A) 2x – y + 7 = 0 (B) 2x + y – 7 = 0 (C) 2x + y + 7 = 0 (D) 2x – y – 7 = 0
5. The area of the triangle formed by the tangents from the points (h, k) to the circle x2 + y2 = a2
and the line joining their points of contact is -

(h 2 + k 2 - a 2 )3/2 (h 2 + k 2 - a 2 )1/2 (h 2 + k 2 - a 2 )3/2 (h 2 + k 2 - a 2 )1/2

(A) a (B) a (C) (D)
h2 + k 2 h2 + k2 h2 + k 2 h2 + k2
6. Tangents drawn from origin to the circle x2 + y2 – 2ax – 2by + b2 = 0 are perpendicular to each
other, if -
(A) a – b = 1 (B) a + b = 1 (C) a2 = b2 (D) a2 + b2 = 1
7. In a system of three curves C1, C2 and C3. C1 is a circle whose equation is x2 + y2 = 4. C2 is the
locus of orthogonal tangents drawn on C1. C3 is the intersection of perpendicular tangents
drawn on C2.
Area enclosed between the curve C2 and C3 is-
(A) 8p sq. units (B) 16p sq. units (C) 32p sq. units (D) None of these
8. Of the two concentric circles the smaller one has the equation x2 + y2 = 4. If each of the two
intercepts on the line x + y = 2 made between the two circles is 1, the equation of the larger
circle is -

(A) x2 + y2 = 5 (B) x2 + y2 = 5 + 2 2 (C) x2 + y2 = 7 + 2 2 (D) x2 + y2 = 11

COMPREHENSION CORRECT TYPE
Paragraph for Question 9 & 10
If x2 + y2 = k2, (where x and y are variables and k is constant), then x & y can be taken
as x = kcosq & y = ksinq.
9. If (x – 1)2 + (y – 2)2 = 4, then maximum value of x2 + y2 is equal to -

(A) 9 - 4 5 (B) 5 + 2 3 (C) 9 + 4 5 (D) 3 + 4 5

9 9
10. If x2 + y2 = 9, then minimum value of the expression 2
+ 2 is equal to -
x y

(A) 1 (B) 2 (C) 3 (D) 4

8
 SUBJECTIV TYPE

11. Let A (– 4, 0) and B (4, 0). Number of points C = (x, y) on the circle x 2 + y2 = 16 such that
the area of the triangle whose vertices are A, B and C is a positive integer, is
12. Determine the nature of the quadrilateral formed by four lines 3x + 4y – 5 = 0; 4x – 3y – 5 = 0;
3x + 4y + 5 = 0 and 4x – 3y + 5 = 0 . Find the equation of the circle inscribed and circum-
scribing this quadrilateral.

9
RACE # 86 CIRCLE

SINGLE CORRECT TYPE

1. The equation of the circle passing through the point (1, 1) and through the point of intersec-
tion of circles x2 + y2 + 13x – 3y = 0 and 2x2 + 2y2 + 4x – 7y – 25 = 0 is-
(A) 4x2 + 4y2 – 17x – 10y + 25 = 0 (B) 4x2 + 4y2 + 30x – 13y – 25 = 0
(C) 4x2 + 4y2 – 30x – 10y – 25 = 0 (D) None of these
2. One possible equation of the chord of x2 + y2 = 100 that passes through (1, 7) and subtends an
2p
angle at origin is -
3
(A) 3y + 4x – 25 = 0 (B) x + y – 8 = 0
(C) 3x + 4y – 31 = 0 (D) None of these
3. The angle between tangents drawn from a point P to the circle x2 + y2 + 4x – 2y – 4 = 0 is 60°.
Then locus of P is -
(A) x2 + y2 + 4x – 2y – 31 = 0 (B) x2 + y2 + 4x – 2y – 21 = 0
(C) x2 + y2 + 4x – 2y – 11 = 0 (D) x2 + y2 + 4x – 2y = 0
4. 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
5. Three equal circles each of radius r touch one another. The radius of the circle touching all
the three given circle internally is

(
(A) 2 + 3 r ) (B)
(2 + 3 ) r (C)
(2 - 3 ) r (
(D) 2 - 3 r )
3 3

6. 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

AB.AD AB . AD AB.AD
(A) (B) (C) AB . AD (D)
2
AB + AD 2 AB + AD AB 2 - AD 2

7. Let P(–1, 0), Q(0, 0) and R(3, 3 3 ) be three points. Then the equation of the bisector of the
angle PQR is

3 3
(A) x+y=0 (B) x + 3y=0 (C) 3x+y=0 (D) x + y=0
2 2
8. lx + my + n = 0 is a tangent line to the circle x2 + y2 = r2 , if
(A) l2 + m2 = n2 r2 (B) l2 + m2 = n2 + r2
(C) n2 = r2(l2 + m2) (D) none of these
9. The greatest distance of the point P(10, 7) from the circle x2 + y2 – 4x – 2y – 20 = 0 is
(A) 5 (B) 15 (C) 10 (D) 17
10. A straight line through the origin O meets the parallel lines 4x + 2y = 9 and 2x + y + 6 = 0 at
points P and Q respectively. Then the point O divides the segment PQ in the ratio
(A) 1 : 2 (B) 3 : 4 (C) 2 : 1 (D) 4 : 3
10
 COMPREHENSION TYPE
(Question 8 to 9)
To the circle x + y = 4 two tangents are drawn from P(–4, 0), which touches the circle at A
2 2

and B and a rhombus PA P' B is completed.

On the basis of above passage, answer the following questions :
11. Circumcentre of the triangle PAB is at
æ 3 ö
(A) (–2, 0) (B) (2, 0) (C) çç 2 , 0 ÷÷ (D) None of these
è ø
12. Ratio of the area of triangle PAB to that of P' AB is
(A) 2 : 1 (B) 1 : 2 (C) 3 : 2 (D) 1 : 1

11
RACE # 87 CIRCLE

SINGLE CORRECT TYPE

1. The area bounded by the angle bisectors of the lines x2 – y2 + 2y = 1 and the line x + y = 3, is
(A) 2 (B) 3 (C) 4 (D) 6
2. A line meets the co-ordinate axes in A and B. A circle is circumscribed about the triangle
OAB.If d1 and 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 :

2d1 + d2 d1 + 2d2 d1d2

(A) (B) (C) d1 + d2 (D) d + d
2 2 1 2
2 2
3. If x + y = 4x + 4y , then all possible values of (x – y) is given by
(A) (–4, 4) (B) {– 4, 4} (C) [– 4, 4] (D) [– 2, 2]
4. Tangents are drawn from (4, 4) to the circle x 2 + 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 (D) 6 2
5. Pair of tangents are drawn from every point on the line 3x + 4y = 12 on the circle x2 + y2 = 4.
Their variable chord of contact always passes through a fixed point whose co-ordinates are
æ4 3ö æ3 3ö æ 4ö
(A) ç 3 , 4 ÷ (B) ç 4 , 4 ÷ (C) (1, 1) (D) ç 1, 3 ÷
è ø è ø è ø
6. The locus of the mid-points of the chords of the circle x2 + y2 – 2x – 4y – 11 = 0 which subtend
60º at the centre is
(A) x2 + y2 – 4x – 2y – 7 = 0 (B) x2 + y2 + 4x + 2y – 7 = 0
(C) x2 + y2 – 2x – 4y – 7 = 0 (D) x2 + y2 + 2x + 4y + 7 = 0
7. The locus of the centres 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
8. 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
(C) x2 + y2 + 2x + 4y – 13 = 0 (D) none
9. AB is a diameter of a circle. CD is a chord parallel to AB and 2CD = AB. The tangent at B
meets the line AC produced at E then AE is equal to
(A) AB (B) 2 AB (C) 2 2 AB (D) 2AB
10. A variable circle passes through the fixed point A(p, q) and touches x-axis. The locus of the
other end of the diameter through A is -
(A) (x – p)2 = 4qy (B) (x – q)2 = 4py
(C) (y – p)2 = 4qy (D) (y – q)2 = 4py
x
11. If (a, a2) falls inside the angle made by the lines y = , x > 0 and y = 3x, x > 0, then a
2
belongs to
æ1 ö æ 1ö æ 1ö
(A) (3, ¥) (B) ç 2 ,3 ÷ (C) ç - 3,- 2 ÷ (D) ç 0, 2 ÷
è ø è ø è ø
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RACE # 88 CIRCLE
SINGLE CORRECT TYPE
1. Consider the circle x2 + (y – 1)2 = 9, (x – 1)2 + y2 = 25. They are such that-
(A) each of these circles lies outside the other
(B) one of these circles lies entirely inside the other
(C) these circles touch each other
(D) they intersect in two points
2. If the circles x2 + y2 + 2x – 8y + 8 = 0 and x2 + y2 + 10x – 2y + 22 = 0 touch each other, their point
of contact is-

æ 17 11 ö æ 11 ö æ 17 11 ö æ 11 ö
(A) ç - , ÷ (B) ç ,2 ÷ (C) ç , ÷ (D) ç - ,2 ÷
è 5 5 ø è 3 ø è 5 5 ø è 3 ø
3. The common chord of x2 + y2 – 4x – 4y = 0 and x2 + y2 = 16 subtends at the origin an angle equal
to-
(A) p/6 (B) p/4 (C) p/3 (D) p/2
4. The distance from the centre of the circle x2 + y2 = 2x to the straight line passing through the
points of intersection of the two circles x2 + y2 + 5x – 8y + 1 = 0, x2 + y2 – 3x + 7y – 25 = 0 is-
(A) 1 (B) 2 (C) 3 (D) None of these
5. The length of the common chord of the circle x2 + y2 + 4x + 6y + 4 = 0 and x2 + y2 + 6x + 4y + 4 = 0
is-

(A) 10 (B) 22 (C) 34 (D) 38

6. The circle x2 + y2 + 2gx + 2fy + c = 0 bisects the circumference of the circle x2 + y2 + 2ax + 2by + d = 0,
then -
(A) 2a (g – a) + 2b (f – b) = c – d (B) 2a (g + a) + 2b (f + b) = c + d
(C) 2g (g – a) + 2f (f – b) = d – c (D) 2g (g + a) + 2f (f + b) = c + d
7. The circles whose equations are x2 + y2 + c2 = 2ax and x2 + y2 + c2 – 2by = 0 will touch one
another externally if -

1 1 1 1 1 1 1 1 1
(A) 2 + 2 = 2 (B) 2 + 2 = 2 (C) 2 + 2 = (D) None of these
b c a c a b a b c2
8. Consider four circles (x ± 1)2 + (y ± 1)2 = 1, then the equation of smaller circle touching these
four circle is

(A) x2 + y2 = 3 – 2 (B) x2 + y2 = 6 – 3 2 (C) x2 + y2 = 5 – 2 2 (D) x2 + y2 = 3 – 2 2

9. The locus of the centre of a circle of radius 2 which rolls on the outside of the circle x2 + y2 +
3x – 6y – 9 = 0 is
(A) x2 + y2 + 3x – 6y + 5 = 0 (B) x2 + y2 + 3x – 6y – 31 = 0

29
(C) x2 + y2 + 3x – 6y + =0 (D) x2 + y2 + 3x – 6y – 5 = 0
4

13
 MULTIPLE CORRECT TYPE
10. Chord AB of the circle x2 + y2 = 100 passes through the point (7, 1) and subtends an angle
of 60° at the circumference of the circle. If m1 and m2 are the slopes of two such chords
then the value of m1m2, is not equal to
(A) – 1 (B) 1 (C) 7/12 (D) – 3
11. Slope of tangent to the circle (x – r) 2 + y2 = r2 at the point (x, y) lying on the circle is

x r-x y2 - x2 y2 + x2
(A) y - r (B) y
(C) 2xy
(D) 2xy

12. The centre(s) of the circle(s) passing through the points (0, 0), (1, 0) and touch-
ing the circle x 2 + y 2 = 9 is/are

æ 3 1ö æ1 3ö æ1 1/ 2 ö æ1 1/ 2ö
(A) ç 2 , 2 ÷ (B) ç 2 , 2 ÷ (C) ç 2 , 2 ÷ (D) ç 2 , - 2 ÷
è ø è ø è ø è ø

14
RACE # 89 CIRCLE
SINGLE CORRECT TYPE
1. The equation of the circle whose diameter is the common chord of the circles
x2 + y2 + 3x +2y + 1 = 0 and x2 + y2 + 3x + 4y + 2 = 0 is-
(A) x2 + y2 + 3x + y + 5 = 0
(B) x2 + y2 + x + 3y + 7 = 0
(C) x2 + y2 + 2x +3 y + 1 = 0
(D) 2(x2 + y2) + 6x + 2y + 1 = 0
2. The equation of the radical axis of circles x2+ y2 + x – y + 2 = 0 & 3x2 + 3y2 – 4x – 12 = 0 is –
(A) 5x – y + 14 = 0
(B) 2x2 + 2y2 – 5x + y – 14 = 0
(C) 7x – 3y + 18 = 0
(D) None of these
3. If the two circles (x – 1)2 + (y – 3)2 = r2 and x2 + y2 – 8x + 2y + 8 = 0 intersect in two distinct
points then -
(A) 2 < r < 8 (B) r < 2 (C) r = 2, r = 8 (D) r > 2
4. The distance between the chords of contact of the tangents to the circle x2 + y2 + 2gx + 2fy + c = 0
from the origin and from the point (g, f) is -

2 2 2 2
1 g +f +c 1 g +f -c
(A) g + f
2 2
(B) (g + f + c)
2 2
(C) (D)
2 g +f
2 2
2 g +f
2 2

5. Consider the figure and find radius of bigger circle. C1 is centre of bigger circle and radius of
smaller circle is unity-

30° ( • C•2
C1

(A) 1 + 2 – 6 (B) 2 + 3

(C) –1 + 2 + 6 (D) 1 + 2 + 6
6. A circle with centre A and radius 7 is tangent to the sides of an angle of 60°. A larger circle
with centre B is tangent to the sides of the angle and to the first circle. The radius of the
larger circle is
(A) 30 (B) 21 (C) 20 (D) 30

15
 COMPREHENSION CORRECT TYPE
Passage for question 7 to 9
Let C1, C2 are two circles each of radius 1 touching internally the sides of triangles POA1,
PA1A2 respectively where Pº (0, 4) O is origin, A1, A2 are the points on positive x-axis.
On the basis of above passage, answer the following questions.
7. Angle subtended by circle C1 at P is-

2 2 3 3
(A) tan–1 (B) 2 tan–1 (C) tan–1 (D) 2 tan–1
3 3 4 4
8. Centre of circle C2 is-

1 3
(A) (3, 1) (B) (3 , 1) (C) (3 , 1) (D) None of these
2 4
9. Length of tangent from P to circle C2 :-

9 19
(A) 4 (B) (C) 5 (D)
2 4

16
RACE # 90 CIRCLE

SINGLE CORRECT TYPE

1. If both roots of x2 + px + q = 0 are positive and one root is cube of other root, then -
(A) q3 – 2q2 – p3 + 4p + q = 0 (B) q3 – 2q2 – p4 + 4p2q + q = 0
3 2 2 3 2 3
(C) q – 2q – p – 4pq + q = 0 (D) q – 2q – p – 4pq + q = 0

2. For x Î [1,16], M and m denotes maximum and minimum values of ƒ ( x ) = log 22 x - log 2 x3 + 3
respectively, then value of (2M – 4m) is-
(A) 5 (B) 2 (C) 8 (D) 11
3. If tan25º + tan85º + tan145º = a + b c , where a,b,c Î I and c is prime number, then value
of (a + b + c) is-
(A) 12 (B) 5 (C) 6 (D) 10
COMPREHENSTION TYPE
Paragraph for Question 4 & 5
S
P
B A'
20 cm 80 cm

O O'
A B'
Q
driver pulley R
driven pulley

A belt drive system is given in the figure. Such that distance between centres (OO') = 120
cm and radii are 20 cm and 80 cm respectively.
4. Length of the belt PQRS is equal to -

(A) 120( p + 3)cm (B) 120( p - 3)cm

(C) 60(2p + 3)cm (D) 60(2p - 3)cm

5. Length of the internal common tangent (AA') is equal to -

(A) 44 10 cm (B) 10 44 cm (C) 22 20 cm (D) 20 22 cm

Paragraph for Question 6 to 8
Two circles S1 : x2 + y2 – 2x – 2y – 7 = 0 and S2 : x2 + y2 – 4x – 4y – 1 = 0 intersects
each other at A and B. On the basis of above passage, answer the following questions:
6. Length of AB is-

(A) 6 (B) 33 (C) 34 (D) 35

17
7. Equation of circle passing through A and B whose AB is diameter-
(A) x2 + y2 – 3x – 3y – 5 = 0 (B) x2 + y2 – 3x – 3y – 4 = 0
(C) x2 + y2 + 3x + 3y – 4 = 0 (D) x2 + y2 + 3x + 3y – 5 = 0
8. Mid point of AB is-

æ5 1ö æ3 3ö
(A) ç 2 , 2 ÷ (B) ç 2 , 2 ÷ (C) (2, 1 (D) (1, 2)
è ø è ø

MATRIX MATCH
Q.9 has four statements (A,B,C & D) given in Column-I and four statements (P, Q, R & S) given
in Column-II. Any given statement in Column-I can have correct matching with one or
more statement(s) given in Column-II.
9. In DPQR, PQ = 3, PR = 4, QR = 5, if internal angle bisector through point P intersect
side QR at S and ÐPQR = a.
Column-I Column-II
QS a
(A) If = where a & b are coprime numbers, (P) 7
SR b
then value of |2b – a| is
(B) Radius of circumcircle of DPQR is k then 2k is (Q) 5
(C) Radius of incircle of DPQR is (R) 2
m
(D) Value of sin2a is where m and n are coprime numbers, (S) 1
n

then value of 4m - 3n is
3

18
 ANSWER KEY
RACE # 80
1. A 2. B 3. B 4. D 5. A 6. B
7. B 8. C 9. 1 10. 5

æ5 ö 1 æ 3ö
11. (a) ç ,1 ÷ ; 13 ; (b) ç 2, ÷ , 1 12. (a) (– 2, – 7); (b) (– 1, 1)
è6 ø 6 è 2ø

RACE # 81
1. B 2. A 3. D 4. D 5. C 6. B,C,D
7. x + y – 10x – 10y + 25 = 0
2 2

8. x2 + y2 – 6x + 4y + 9 = 0, or x2 + y2 + 10x + 20y + 25 = 0
9. x2 + y2 – 6x + 2y – 15 = 0
10. 3 11. A-S, B-P, C-R,T, D-S
RACE # 82
1. B 2. C 3. D 4. D 5. B 6. B
7. B 8. C 9. B
RACE # 83
1. A 2. B 3. B 4. C 5. D 6. B
7. B 8. D 9. B 10. A 11. C 12. A
RACE # 84
1. C 2. D 3. B 4. D 5. A 6. B
7. A 8. A 9. A 10. C 11. C
RACE # 85
1. A 2. A 3. C 4. B 5. A 6. C
7. A 8. B 9. C 10. D 11. 62
12. Square of side 2, x + y = 1, x2 + y2 = 2
2 2

RACE # 86
1. B 2. A 3. A 4. A 5. B 6. D
7. C 8. C 9. B 10. B 11. A 12. D
RACE # 87
1. A 2. C 3. C 4. B 5. D 6. C
7. A 8. A 9. D 10. A 11. B
RACE # 88
1. B 2. A 3. D 4. B 5. C 6. A
7. C 8. D 9. B 10. B,C,D 11. B,C 12. C,D
RACE # 89
1. D 2. C 3. A 4. D 5. D 6. B
7. C 8. B 9. B
RACE # 90
1. B 2. D 3. A 4. A 5. B 6. C
7. B 8. B 9. A-R, B-Q, C-S, D-P

19