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

2. If the circumference of the circle $x^2+y^2+8x+8y-b=0$ is bisected by the circle $x^2+y^2-2x+4y+a=0$ 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 $x^2+y^2-4x-2y=8$ and $x^2+y^2-2x-4y=8$ and the point $(-1,4)$ is

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

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

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

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

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 $x^2+y^2-4x-5=0$ and $x^2+y^2+6x-2y+6=0$ 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 $x^2+y^2+2x+2ky+6=0$ and $x^2+y^2+2ky+k=0$ 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 $x^2+y^2+2gx+2fy+c=0$ from the origin and the point $(\mathrm{g},\mathrm{f})$ is

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

(b) ${\displaystyle \frac{1}{2}}({\mathrm{g}}^2+{\mathrm{f}}^2+\mathrm{c})$

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

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

EXERCISE:

1. Let $0<\alpha <{\displaystyle \frac{\pi }{2}}$ 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 $x^2+y^2+6x+6y=2$ meets the straight line $5x-2y+6=0$ 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 $\mathrm{△}\mathrm{ABC}$ 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 $x^2+y^2-6x-8y=0$ are perpendicular to each other is

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

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

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

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

6. If the two circles $x^2+y^2+2gx+2fy=0$ and $x^2+y^2+2g_{1}x+2f_{1}y=0$ 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 $x2+y2+\lambda x+(1-\lambda )y+5=0$ 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 $x^2+y^2+4x-7y+12=0$ cuts an intercept on $y$-axis of length

(a) 3

(b) 4

(c) 7

(d) 1

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

(a) $x+y=0$

(b) $x+3y=0$

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

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

10. The coordinates of the middle point of the chord cut off by $2x-5y+18=0$ by the circle $x^2+y^2-6x+2y-54=0$ 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 $\mathrm{\forall }a\in (-\mathrm{\infty },0)$ and $P$ moves on a plane such that $\mathrm{PA}=\mathrm{nPB}(\mathrm{n}\ne 0)$.

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

11. If $|\mathrm{n}|\ne 1$, 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 $\mathrm{A}$