1. Find the equation of the system of circles co-axial with the circles $x^{2}+y^{2}+4 x+2 y+1=0$ and $x^{2}+y^{2}-$ $2 x+6 y-6=0$. Also find the equation of that particular circles whose centre lies on radical axis.

2. If the circumference of the circle $x^{2}+y^{2}+8 x+8 y-b=0$ is bisected by the circle $x^{2}+y^{2}-2 x+4 y+a=0$ then $a+b$ equal to

(a) 50

(b) 56

(c) -56

(d) -34

3. The equation of the circle passing through the point of intersection of the circles $x^{2}+y^{2}-4 x-2 y=8$ and $x^{2}+y^{2}-2 x-4 y=8$ and the point $(-1,4)$ is

(a) $x^{2}+y^{2}+4 x+4 y-8=0$

(b) $x^{2}+y^{2}-3 x+4 y+8=0$

(c) $x^{2}+y^{2}+x+y-8=0$

(d) $x^{2}+y^{2}-3 x-3 y-8=0$

4. If the common chord of the circles $x^{2}+(y-b)^{2}=16$ and $x^{2}+y^{2}=16$ subtends a right angle at the origin then $b=$

(a) 4

(b) $4 \sqrt{2}$

(c) $-4 \sqrt{2}$

(d) 8

5. Given the circles $x^{2}+y^{2}-4 x-5=0$ and $x^{2}+y^{2}+6 x-2 y+6=0$ Let $P$ be a point $(\alpha, \beta)$ such that the tangents from $\mathrm{P}$ to both the circles are equal. Then

(a) $2 \alpha+10 \beta+11=0$

(b) $2 \alpha-10 \beta+11=0$

(c) $10 \alpha-2 \beta+11=0$

(d) $10 \alpha+2 \beta+11=0$

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

(a) 2 or $-3 / 2$

(b) $-2 \mathrm{or}-3 / 2$

(c) 2 or $3 / 2$

(d) $-2 \mathrm{or}3 / 2$

7. The distance between the chords of contact of the tangents to the circle $x^{2}+y^{2}+2 g x+2 f y+c=0$ from the origin and the point $(\mathrm{g}, \mathrm{f})$ is

(a) $\mathrm{g}^{2}+\mathrm{f}^{2}$

(b) $\dfrac{1}{2}\left(\mathrm{~g}^{2}+\mathrm{f}^{2}+\mathrm{c}\right)$

(c) $\dfrac{1}{2}\left(\dfrac{\mathrm{g}^{2}+\mathrm{f}^{2}+\mathrm{c}}{\sqrt{\mathrm{g}^{2}+\mathrm{f}^{2}}}\right)$

(d) $\dfrac{1}{2} \dfrac{\mathrm{g}^{2}+\mathrm{f}^{2}-\mathrm{c}}{\sqrt{\mathrm{g}^{2}+\mathrm{f}^{2}}}$

EXERCISE:

1. Let $0<\alpha<\dfrac{\pi}{2}$ be a fixed angle. If $\mathrm{P}=(\cos \theta, \sin \theta)$ and $\mathrm{Q}=\cos (\alpha-\theta), \sin (\alpha-\theta) \mathrm{Q}$ obtained form $\mathrm{P}$ by

(a) clockwise rotation around origin through an angle $\alpha$

(b) anti-clockwise rotation around origin through an angle $\alpha$

(c) reflection in the line through origin with slope $\tan \alpha$

(d) reflection in the line through origin with slope $\tan \alpha / 2$

2. 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 $\mathrm{Q}$ on the $\mathrm{y}$-axis, then the length of $\mathrm{PQ}$ is

(a) 4

(b) $2 \sqrt{5}$

(c) 5

(d) $3 \sqrt{5}$

3. The equations to the sides $\mathrm{AB}, \mathrm{BC}, \mathrm{CA}$ of a $\triangle \mathrm{ABC}$ are drawn on $\mathrm{AB}, \mathrm{BC}, \mathrm{CA}$ as diameters. The point of concurrence of the common chord is

(a) centroid of the triangle

(b) orthocenter

(c) circumcentre

(d) incentre

4. The number of rational points ( $a$ point $(a, b)$ is rational, if $a$ and $b$ both are rational numbers) on the circumference of a circle having centre $(\pi, \mathrm{e})$ is

(a) at most one

(b) at least two

(c) exactly two

(d) infinite

5. The locus of a point such that the tangents drawn from it to the circle $x^{2}+y^{2}-6 x-8 y=0$ are perpendicular to each other is

(a) $\mathrm{x}^{2}+\mathrm{y}^{2}-6 \mathrm{x}-8 \mathrm{y}-25=0$

(b) $x^{2}+y^{2}+6 x-8 y-5=0$

(c) $x^{2}+y^{2}-6 x+8 y-5=0$

(d) $x^{2}+y^{2}-6 x-8 y+25=0$

6. If the two circles $x^{2}+y^{2}+2 g x+2 f y=0$ and $x^{2}+y^{2}+2 g _{1} x+2 f _{1} y=0$ touch each other, then

(a) $f _{1} g=f g _{1}$

(b) $\mathrm{ff} _{1}=\mathrm{gg} _{1}$

(c) $\mathrm{f}^{2}+\mathrm{g}^{2}=\mathrm{f} _{1}^{2}+\mathrm{g} _{1}^{2}$

(d) none of these

7. The number of integral values of $\lambda$ for which $x 2+y 2+\lambda x+(1-\lambda) y+5=0$ is the equation of a circle whose radius cannot exceed 5 , is

(a) 14

(b) 18

(c) 16

(d) none of these

8. The circle $x^{2}+y^{2}+4 x-7 y+12=0$ cuts an intercept on $y$-axis of length

(a) 3

(b) 4

(c) 7

(d) 1

9. One of the diameter of the circle $x^{2}+y^{2}-12 x+4 y+6=0$ is given by

(a) $x+y=0$

(b) $x+3 y=0$

(c) $\mathrm{x}=\mathrm{y}$

(d) $3 x+2 y=0$

10. The coordinates of the middle point of the chord cut off by $2 x-5 y+18=0$ by the circle $x^{2}+y^{2}-6 x+2 y-54=0$ are

(a) $(1,4)$

(b) $(2,4)$

(c) $(4,1)$

(d) $(1,1)$

PASSAGE - 1

Let $A \equiv(a, 0)$ and $B \equiv(-a, 0)$ be two fixed points $\forall a \in(-\infty, 0)$ and $P$ moves on a plane such that $\mathrm{PA}=\mathrm{nPB}(\mathrm{n} \neq 0)$.

On the basis of above information, answer the following questions:

11. If $|\mathrm{n}| \neq 1$, then the locus of a point $\mathrm{P}$ is

(a) a straight line

(b) a circle

(c) a parabola

(d) an ellipse

12. If $\mathrm{n}=1$, then the locus of a point $\mathrm{P}$ is

(a) a straight line

(c) a circle

(c) a parabola

(d) a hyperbola

13. If $0<\mathrm{n}<1$, then

(a) A lies inside the circle and $\mathrm{B}$ lies outside the circle

(b) A lies outside the circle and $\mathrm{B}$ lies inside the circle

(c) both $\mathrm{A}$ and $\mathrm{B}$ lies on the circle

(d) both $A$ and $B$ lies inside the circle

14. If $\mathrm{n}>1$, then

(a) A lies outside the circle and $\mathrm{B}$ lies inside the circle

(b) A lies outside the circle and $B$ lies inside the circle

(c) both $\mathrm{A}$ and $\mathrm{B}$ lies on the circle

(d) both $\mathrm{A}$ and $\mathrm{B}$ lies inside the circle

15. If focus of $\mathrm{P}$ is a circle, then the circle

(a) passes through $\mathrm{A}$ and $\mathrm{B}$

(b) never passes through $A$ and $B$

(c) passes through $A$ but does not pass through $B$

(d) passes through $\mathrm{B}$ but does not pass through $\mathrm{A}$