If the point of intersection of the tangents drawn at the points where the line $5x + y + 1 = 0$ cuts the circle $x^2 + y^2 - 2x - 6y - 8 = 0$ is $(a, b)$,then $5a + b =$

Tangents are drawn from the point $(-1, -4)$ to the circle $x^2 + y^2 - 2x + 4y + 1 = 0$. The length of the corresponding chord of contact is:

Two circles centered at $(2,3)$ and $(4,5)$ intersect each other. If their radii are equal,then the equation of the common chord is

$A$ point that lies on the common tangent to the circles $x^2+y^2-2x+18y+78=0$ and $x^2+y^2+8x-6y-200=0$ among the following options is

The length of the common chord of the circles $x^2 + y^2 - 6x - 16 = 0$ and $x^2 + y^2 - 8y - 9 = 0$ is:

Let $C_{1}$ and $C_{2}$ denote the centres of the circles $x^{2}+y^{2}=4$ and $(x-2)^{2}+y^{2}=1$ respectively and let $P$ and $Q$ be their points of intersection. Then, the areas of $\Delta C_{1} P Q$ and $\Delta C_{2} P Q$ are in the ratio (in $: 1$)