The number of common tangents to the circles $x^{2}+y^{2}=4$ and $x^{2}+y^{2}-6x-8y-24=0$ is

If $r_1$ and $r_2$ are the radii of two circles that touch all four circles given by $(x \pm r)^2 + (y \pm r)^2 = r^2$,then find the value of $\frac{r_1+r_2}{r}$.

The position of the point $(1, 1)$ with respect to the circle $x^2 + y^2 - x + y - 1 = 0$ is:

Let a circle $C$ touch the lines $L_{1}: 4x - 3y + K_{1} = 0$ and $L_{2}: 4x - 3y + K_{2} = 0$,where $K_{1}, K_{2} \in R$. If a line passing through the centre of the circle $C$ intersects $L_{1}$ at $(-1, 2)$ and $L_{2}$ at $(3, -6)$,then the equation of the circle $C$ is:

Let the point $B$ be the reflection of the point $A(2,3)$ with respect to the line $8x-6y-23=0$. Let $\Gamma_A$ and $\Gamma_B$ be circles of radii $2$ and $1$ with centres $A$ and $B$ respectively. Let $T$ be a common tangent to the circles $\Gamma_A$ and $\Gamma_B$ such that both the circles are on the same side of $T$. If $C$ is the point of intersection of $T$ and the line passing through $A$ and $B$,then the length of the line segment $AC$ is.

Let the circles $C_1: (x-\alpha)^2 + (y-\beta)^2 = r_1^2$ and $C_2: (x-8)^2 + (y-\frac{15}{2})^2 = r_2^2$ touch each other externally at the point $(6,6)$. If the point $(6,6)$ divides the line segment joining the centres of the circles $C_1$ and $C_2$ internally in the ratio $2:1$,then $(\alpha+\beta) + 4(r_1^2 + r_2^2)$ equals