Two tangents are drawn from a point $P$ to the circle $x^{2}+y^{2}-2 x-4 y+4=0$, such that the angle between these tangents is $\tan ^{-1}\left(\frac{12}{5}\right)$, where $\tan ^{-1}\left(\frac{12}{5}\right) \in(0, \pi)$. If the centre of the circle is denoted by $C$ and these tangents touch the circle at points $A$ and $B$, then the ratio of the areas of $\Delta PAB$ and $\Delta CAB$ is :

Square of the length of the tangent drawn from the point $(\alpha ,\beta )$ to the circle $a{x^2} + a{y^2} = {r^2}$ is

Tangents are drawn from $(4, 4) $ to the circle $x^2 + y^2 – 2x – 2y – 7 = 0$ to meet the circle at $A$ and $B$. The length of the chord $AB $ is

Let a circle $C$ touch the lines $L_{1}: 4 x-3 y+K_{1}$ $=0$ and $L _{2}: 4 x -3 y + K _{2}=0, K _{1}, K _{2} \in R$. 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

Given the circles ${x^2} + {y^2} – 4x – 5 = 0$and ${x^2} + {y^2} + 6x – 2y + 6 = 0$. Let $P$ be a point $(\alpha ,\beta )$such that the tangents from P to both the circles are equal, then