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

The equation of the tangent to the circle at the point $(1, -1)$,whose center is the point of intersection of the straight lines $x - y = 1$ and $2x + y = 3$,is:

The line $y = x + a\sqrt{2}$ is a tangent to the circle $x^2 + y^2 = a^2$ at which of the following points?

Let $ABCD$ be a square of side length $1$. $A$ circle $C_{1}$ centered at $A$ with unit radius is drawn. Another circle $C_{2}$ which touches $C_{1}$ and is tangent to the lines $AD$ and $AB$ is also drawn. Let a tangent line from the point $C$ to the circle $C_{2}$ meet the side $AB$ at $E$. If the length of $EB$ is $\alpha+\sqrt{3} \beta,$ where $\alpha, \beta$ are integers,then $\alpha+\beta$ is equal to.........

The point where the line $4x - 3y + 7 = 0$ touches the circle $x^2 + y^2 - 6x + 4y - 12 = 0$ is

The equation of the pair of tangents drawn from the point $(1, 1)$ to the circle $x^2 + y^2 + 2x + 2y + 1 = 0$ is: