Let the tan gents drawn to the circle, $x^2 + y^2 = 16$ from the point $P(0, h)$ meet the $x-$ axis at point  $A$ and $B.$ If the area of $\Delta APB$ is minimum, then $h$  is equal to

If the line $lx + my + n = 0$ be a tangent to the circle ${(x – h)^2} + {(y – k)^2} = {a^2},$ then

Tangents are drawn from the point $(4, 3)$ to the circle ${x^2} + {y^2} = 9$. The area of the triangle formed by them and the line joining their points of contact is

The centres of two circles $C_1$ and $C_2$ each of unit radius are at a distance of $6$ units from each other. Let $P$ be the mid point of the line segment joining the centres of $C_1$ and $C_2$ and $C$ be a circle touching circles $C_1$ and $C_2$ externally. If a common tangent to $C_1$ and $C$ passing through $P$ is also a common tangent to $C_2$ and $C$, then the radius of the circle $C$ is

Point $M$ moved along the circle $(x – 4)^2 + (y – 8)^2 = 20 $. Then it broke away from it and moving along a tangent to the circle, cuts the $x-$ axis at the point $(- 2, 0)$ . The co-ordinates of the point on the circle at which the moving point broke away can be :