The equation of the chord of contact,if the tangents are drawn from the point $(5, -3)$ to the circle $x^2 + y^2 = 10$,is

The area of the triangle (in sq. units) formed by the tangents drawn from $P(4,4)$ to the circle $S \equiv x^2+y^2-2x-2y-7=0$ and the chord of contact of $P$ with respect to $S=0$ is

Let $P$ and $Q$ be two external points of the circle $S \equiv x^2+y^2-a^2=0$. Let the chord of contact of the point $P$ with respect to the circle $S=0$ pass through $Q$. If $l_1$ and $l_2$ are the lengths of the tangents drawn from $P$ and $Q$ to the circle $S=0$,then $PQ=$

The length of the common chord of the circles of radii $15$ and $20$,whose centers are $25$ units of distance apart,is

Consider two circles $C_1: x^2+y^2=25$ and $C_2: (x-\alpha)^2+y^2=16$,where $\alpha \in (5, 9)$. Let the angle between the two radii (one to each circle) drawn from one of the intersection points of $C_1$ and $C_2$ be $\sin^{-1}\left(\frac{\sqrt{63}}{8}\right)$. If the length of the common chord of $C_1$ and $C_2$ is $\beta$,then the value of $(\alpha \beta)^2$ equals:

If the circles $x^2 + y^2 + 2ax + cy + a = 0$ and $x^2 + y^2 - 3ax + dy - 1 = 0$ intersect at two distinct points $P$ and $Q$,then for what value of $a$ does the line $5x + 6y - a = 0$ pass through points $P$ and $Q$?