RACE # 52 C IR CL E MATH EM ATI CS

1. As shown in the figure, three circles which have the same radius r, have centres at (0,0) ; (1,1) and (2,1). If they
have a common tangent line, as shown then, their radius 'r' is -
y
5 -1 5 C1 r
(A) (B) t r
C2
2 10
r
1 3 -1 O
C 1 2
(C) (D)
2 2
2. Two circles of radii 4 cms and 1 cm touch each externally and q is the angle contained by their direct
common tangents. Then sinq =
(A) 24/25 (B) 12/25 (C) 3/4 (D) none
3. The number of common tangents of the circle (x + 2) + (y – 2) = 49 and (x – 2)2 + (y + 1)2 = 4 is :
2 2

(A) 0 (B) 1 (C) 2 (D) 3

4. If a circle passes through P(0,1), Q(0,9) and touches the x-axis, then which of the following statement(s) is/are
TRUE ?
(A) Centres of circles are (±3, ±5) (B) Equation of one of their direct common tangent is y = 10
(C) Radii of both the circle is 5. (D) length of common chord of circles is 8.
5. Triangle ABC is right angled at A. The circle with centre A and radius AB cuts BC and AC internally at D and E
respectively. If BD = 20 and DC = 16 then the length AC equals

(A) 6 21 (B) 6 26 (C) 30 (D) 32

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 equal to
AB.AD AB.AD AB.AD
(A) (B) (C) AB.AD (D)
AB 2 + AD 2 AB + AD AB 2 - AD 2
7. Let ABC be a triangle with ÐA = 45°. Let P be a point on the side BC with PB = 3 and PC = 5. If 'O' is the
circumcentre of the triangle ABC then the length OP is equal to -

(A) 15 (B) 17 (C) 18 (D) 19

8. The value of 'c' for which the set, {(x, y) | x + y + 2x £ 1} Ç {(x,y) | x – y + c ³ 0} contains only one point in
2 2

common is :
(A) (–¥, –1] È [3, ¥) (B) {–1, 3} (C) {–3} (D) {–1}
9. Locus of all point P(x, y) satisfying x3 + y3 + 3xy = 1 consists of union of
(A) a line and an isolated point (B) a line pair and an isolated point
(C) a line and a circle (D) a circle and a isolated point.
10. Chord AB of the circle x + y = 100 passes through the point (7, 1) and subtends an angle of 60° at the
2 2

circumference of the circle. If m1 and m2 are the slopes of two such chords then the value of m1m2, is
(A) –1 (B) 1 (C) 7/12 (D) –3
11. 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)
12. Circle C1 has radius 2 and circle C2 has radius 3, and the distance between the centers of C1 and C2 is 7. If two
lines, one tangent to both circles and the other passing through the center of both circles, intersect at point P
which lies between the centers of C1 and C2, then the distance between P and the center of C1 is-

9 7 8 14
(A) (B) (C) (D)
4 3 3 5
13. The locus of the point from which two given unequal circle subtend equal angle is :
(A) a straight line (B) a circle (C) a parabola (D) none
14. Locus of the intersection of the two straight lines passing through (1,0) and (–1,0) respectively and including
an angle of 45° can be a circle with -

(A) centre (1,0) and radius 2 (B) centre (1,0) and radius 2.

(C) centre (0,1) and radius 2. (D) centre (0, –1) and radius 2.
[SUBJECTIVE]
15. Find the radical centre of the following set of circles
x2 + y2 – 3x – 6y + 14 = 0; x2 + y2 – x – 4y + 8 = 0; x2 + y2 + 2x – 6y + 9 = 0
16. Find the equation to the circle orthogonal to the two circles
x2 + y2 – 4x – 6y + 11 = 0; x2 + y2 – 10x – 4y + 21 = 0 and has 2x + 3y = 7 as diameter.
17. Find the equation to the circle, cutting orthogonally each of the following circles :
x2 + y2 – 2x + 3y – 7 = 0; x2 + y2 + 5x – 5y + 9 = 0; x2 + y2 + 7x – 9y + 29 = 0.
18. Given that x2 + y2 = 14x + 6y + 6, find the largest possible value of the expression E = 3x + 4y.

y
19. If M and m are the maximum and minimum values of for pair of real number (x,y) which satisfy the
x
equation (x – 3)2 + (y – 3)2 = 6, then find the value of (M + m).
Paragraph for question Nos. 20 to 23
Consider the circle S : x2 + y2 – 4x – 1 = 0 and the line L : y = 3x – 1. If the line L cuts the circle at A and B then
20. Length of the chord AB equal

(A) 2 5 (B) 5 (C) 5 2 (D) 10

21. The angle subtended by the chord AB in the minor arc of S is

3p 5p 2p p
(A) (B) (C) (D)
4 6 3 4
22. Acute angle between the line L and the circle S is

p p p p
(A) (B) (C) (D)
2 3 4 6
23. If the equation of the circle on AB as diameter is of the form x2 + y2 + ax + by + c = 0 then the magnitude of the
r
vector V = aiˆ + bjˆ + ckˆ has the value equal to

(A) 8 (B) 6 (C) 9 (D) 10

 Answers

RACE # 52
1. (B) 2. (A) 3. (B) 4. (B) 5. (B) 6. (D) 7. (B) 8. (D) 9. (A) 10. (A)
11. (D) 12. (D) 13. (B) 14. (CD) 15. (1, 2) 16. x2 + y2 – 4x – 2y + 3 = 0
17. x2 + y2 – 16x – 18y – 4 = 0 18. 73 19. 6 20. (D) 21. (A) 22. (C) 23. (B)