Tangent to the circle $x^2 + y^2$ = $5$ at the point $(1, -2)$ also touches the circle $x^2 + y^2 -8x + 6y + 20$ = $0$ . Then its point of contact is 

The angle between the tangents drawn from the origin to the circle $(x -7)^2 + (y + 1)^2 = 25$ is :-

Which of the following lines is a tangent to the circle ${x^2} + {y^2} = 25$ for all values of $m$.....

A circle $C_{1}$ passes through the origin $O$ and has diameter $4$ on the positive $x$-axis. The line $y =2 x$ gives a chord $OA$ of a circle $C _{1}$. Let $C _{2}$ be the circle with $OA$ as a diameter. If the tangent to $C _{2}$ at the point $A$ meets the $x$-axis at $P$ and $y$-axis at $Q$, then $QA : AP$ is equal to.

Pair of tangents are drawn from every point on the line $3x + 4y = 12$ on the circle $x^2 + y^2 = 4$. Their variable chord of contact always passes through a fixed point whose co-ordinates are

If the tangent at a point $P(x,y)$ of a curve is perpendicular to the line that joins origin with the point $P$, then the curve is