Untitled Document

Question Bank No: 1

1. If one of the diameters of the circle $x^{2} + y^{2} - 2 x - 6 y + 6 = 0$ is a chord to the circle with centre at (2,1), then the radius of the circle is

a)		$\sqrt{3}$
b)		$\sqrt{2}$
c)		3
d)		2

2. The centre of the circle which circumscribes the square formed by $x^{2} - 8 x + 12 = 0$ and $y^{2} - 14 y + 45 = 0$ is

a)		(3, 7)
b)		(4, 7)
c)		(2, 5)
d)		(6, 9)

3. If $a > 2 b > 0$ then the positive value of m for which $y = mx - b \sqrt{1 + m^{2}}$ is a common tangent to $x^{2} + y^{2} = b^{2} and {(x - a)}^{2} + y^{2} = b^{2}$ is

a)		$\frac{2 b}{\sqrt{a^{2} - 4 b^{2}}}$
b)		$\frac{\sqrt{a^{2} - 4 b^{2}}}{2 b}$
c)		$\frac{2 b}{a - 2 b}$
d)		$\frac{b}{a - 2 b}$

4. If the tangent at the point P on the circle $x^{2} + y^{2} + 6 x + 6 y = 2$ meets the straight line $5 x - 2 y + 6 = 0$ at a point Q on the y-axis, then the lengths of PQ is

a)		4
b)		2 $\sqrt{5}$
c)		5
d)		3 $\sqrt{5}$

5. Let PQ and RS be tangents at the extremities of the diameter PR of a circle of radius r. If PS and RQ intersect at a point X on the circumference of the circle, then 2r equals

a)		$\sqrt{PQ \cdot RS}$
b)		$\frac{PQ + RS}{2}$
c)		$\frac{2 PQ \cdot RS}{PQ + RS}$
d)		$\sqrt{\frac{{PQ}^{2} + {RS}^{2}}{2}}$

6. Let AB be a chord of the circle $x^{2} + y^{2} = r^{2}$ subtending a right angle at the centre.Then the locus of the centroid of the triangle PAB as P moves on the circle is

a)		parabola
b)		circle
c)		ellipse
d)		pair of lines

7. If the circles $x^{2} + y^{2} + 2 ky + 6 = 0$ and $x^{2} + y^{2} + 2 ky + k = 0$ intersect orthogonally, then k is

a)		$2 or - \frac{3}{2}$
b)		$- 2 or - \frac{3}{2}$
c)		$2 or \frac{3}{2}$
d)		$- 2 or \frac{3}{2}$

8. The triangle PQR is inscribed in the circle $x^{2} + y^{2} = 25$ . If Q and R have coordinates (3, 4) and (-4, 3) respectively, $∠ QPR$ is

a)		$\frac{π}{2}$
b)		$\frac{π}{3}$
c)		$\frac{π}{4}$
d)		$\frac{π}{6}$

9. Let $L_{1}$ be a line passing through the origin and $L_{2}$ be the line x+y=1. If the intersecpts made by the circles $x^{2} + y^{2} - x + 3 y = 0$ on $L_{1}$ and $L_{2}$ are equal, then $L_{1}$ is

a)		x+y=0
b)		x-y=0
c)		2x+7y=0
d)		x-7y=0

10. If two distinct chords, drawn from the point (p, q) on the circle $x^{2} + y^{2} = px + qy$ , where pq $\neq 0,$ are bisected by the x- axis, then

a)		$p^{2} = q^{2}$
b)		$p^{2} = 8 q^{2}$
c)		$p^{2} < 8 q^{2}$
d)		$p^{2} > 8 q^{2}$

11. The number of common tangents to the circles $x^{2} + y^{2} = 4$ amd $x^{2} + y^{2} - 6 x - 8 y = 24$ is

a)		0
b)		1
c)		3
d)		4

12. Let $A_{0} A_{1} A_{2} A_{3} A_{4} A_{5}$ be a regular hexagon inscribed in a circle of unit radius. Then the product of the lengths of the line segments $A_{0}$ $A_{1}, A_{0} A_{2}$ and $A_{0}$ $A_{4}$ is

a)		$\frac{3}{4}$
b)		3 $\sqrt{3}$
c)		3
d)		$\frac{3 \sqrt{3}}{2}$

13. If the angle between the tangents drawn from P to the circle $x^{2} + y^{2} + 4 x - 6 y + 9 \sin^{2} α + 13 \cos^{2} α = 0$ is 2 $α,$ then the locus of is

a)		$x^{2} + y^{2} + 4 x - 6 y + 14 = 0$
b)		$x^{2} + y^{2} + 4 x - 6 y - 9 = 0$
c)		$x^{2} + y^{2} + 4 x - 6 y - 4 = 0$
d)		$x^{2} + y^{2} + 4 x - 6 y + 9 = 0$

14. The locus of the centre of circle which touches externally the circle $x^{2} + y^{2} - 6 x - 6 y + 14 = 0$ and touches the y-axis is

a)		$x^{2} - 6 x - 10 y + 14 = 0$
b)		$x^{2} - 10 x - 6 y + 14 = 0$
c)		$y^{2} - 6 x - 10 y + 14 = 0$
d)		$y^{2} - 10 x - 6 y + 14 = 0$

15. The centre of a circle passing through the points (0,0) and (1,0) and touching the circle $x^{2} + y^{2} = 9$ is

a)		$(\frac{3}{2}, \frac{1}{2})$
b)		$(\frac{1}{2}, \frac{3}{2})$
c)		$(\frac{1}{2}, \frac{5}{2})$
d)		$(\frac{1}{2}, - \sqrt{2})$

16. A circle passes through the points of intersection of the lines $λ x - y + 1 = 0 and x - 2 y + 3 = 0$ with the coordinate axes, then $λ$ is

a)		0
b)		1
c)		2
d)		$\frac{1}{3}$

17. The lines $2 x - 3 y = 5$ and $3 x - 4 y = 7$ are diameters of a circle of area 154. Taking $π = \frac{22}{7}$ , the equation of the circle is

a)		$x^{2} + y^{2} + 2 x - 2 y = 62$
b)		$x^{2} + y^{2} + 2 x - 2 y = 47$
c)		$x^{2} + y^{2} - 2 x + 2 y = 47$
d)		$x^{2} + y^{2} - 2 x + 2 y = 62$

18. If the circles ${(x - 1)}^{2} + {(y - 3)}^{2} = r^{2}$ and $x^{2} + y^{2} - 8 x + 2 y + 8 = 0$ intersect in two distinct points , then

a)		$2 < r < 8$
b)		$r = 2$
c)		$r < 2$
d)		$r > 2$

19. The equations of tangents drawn from the origin to the circle $x^{2} + y^{2} - 2 r x - 2 h y + h^{2} = 0$ are
cir7

a)		$x = 0$
b)		$(h^{2} + r^{2}) x - 2 r h y = 0$
c)		$y = 0$
d)		$(h^{2} - r^{2}) x + 2 r h y = 0$

20. If a circle passes through the point (a, b) and cuts the circle $x^{2} + y^{2} = k^{2}$ orthogonally, then the locus of its centre is

a)		$2 a x + 2 b y - (a^{2} + b^{2} + k^{2}) = 0$
b)		$2 a x + 2 b y - (a^{2} + b^{2} + k^{2}) = 0$
c)		$x^{2} + y^{2} - 3 a x - 4 b y + a^{2} + b^{2} + k^{2} = 0$
d)		$x^{2} + y^{2} - 2 a x - 3 b y + a^{2} + b^{2} + k^{2} = 0$

21. The number of tangents that can be drawn from the point $(\frac{5}{2}, 1)$ to the circle passing through the points (1, $\sqrt{3}$ ), (1, - $\sqrt{3}$ ) and (3, - $\sqrt{3}$ ) is

a)		1
b)		0
c)		2
d)		none

22. The locus of the mid- points of chords of the circle $x^{2} + y^{2} = 4$ which subtend a right angle at the origin is
cir4

a)		$x + y = 2$
b)		$x^{2} + y^{2} = 1$
c)		$x^{2} + y^{2} = 2$
d)		$x + y = 1$

23. AB is a diameter of a circle and C is any point on the circumference of the circle. Then
cir3

a)		area of $Δ ABC$ is maximum when it is isosceles.
b)		area of $Δ ABC$ is minimum when it is isosceles.
c)		perimeter of $Δ ABC$ is minimum when it is isosceles
d)		none

24. The equation of the circle passing through (1,1) and the points of intersection of the circles $x^{2} + y^{2} + 13 x - 3 y = 0$ and $2 x^{2} + 2 y^{2} + 4 x - 7 y - 25 = 0$ is

a)		$4 x^{2} + 4 y^{2} - 30 x - 10 y - 25 = 0$
b)		$4 x^{2} + 4 y^{2} + 30 x - 13 y - 25 = 0$
c)		$4 x^{2} + 4 y^{2} - 17 x + 10 y + 25 = 0$
d)		none

25. A square is inscribed in the circle $x^{2} + y^{2} - 2 x + 4 y + 3 = 0$ . Its sides are parallel to the coordinate axes. Then one vertex of the square is

a)		( $1 + \sqrt{2}$ , $- 2$ )
b)		( $1 - \sqrt{2}$ , $- 2$ )
c)		( $1$ , $- 2 +$ $\sqrt{2}$ )
d)		none