The length of the common chord of the two circles $2x^{2} + 2y^{2} + 7x - 5y + 2 = 0$ and $x^{2} + y^{2} - 4x + 8y - 18 = 0$ is:

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

The line $3x-y+k=0$ touches the circle $x^2+y^2+4x-6y+3=0$. If $k_1, k_2$ $(k_1 < k_2)$ are the two values of $k$,then the equation of the chord of contact of the point $(k_1, k_2)$ with respect to the given circle is

Which of the following is a point on the common chord of the circles ${x^2 + y^2 + 2x - 3y + 6 = 0}$ and ${x^2 + y^2 + x - 8y - 13 = 0}$?

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: