Director Circle

Q.

Find the equation of director circle of the circle $(x-1{)}^2+(y-2{)}^2=4$

1. $(x-1{)}^2+(y-2{)}^2=4\sqrt{2}$

2. $(x-1{)}^2+(y-2{)}^2=8$

3. $(x+1{)}^2+(y+2{)}^2=4\sqrt{2}$

4. $(x-2{)}^2+(y-1{)}^2=8$

Q. If the line $x-1=0$ is the directrix of the parabola $y^2-kx+8=0,k\ne 0$ and the parabola intersects the circle $x^2+y^2=4$ in two real distinct points, then the value of $k$ is?

1. $-8$
2. $-4$
3. $4$
4. $8$

Q. Consider three circles $C_1,C_2$ and $C_3$ such that $C_2$ is the director circle of $C_1$ and $C_3$ is the director circle of $C_2,$ Tangents to $C_1$ from any point on $C_3$ intersect $C_2$ at $P$ and $Q.$ Find the angle between the tangents to $C_2$ at $P$ and $Q.$ Also identify the locus of the point of intersection of tangents at $P$ and $Q$

Q. A straight line $x=y+2$ touches the circle $4(x^2+y^2)=r^2$. The value of $r$ is:

1. $\sqrt{2}$
2. $2\sqrt{2}$
3. $2$
4. $1$