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 equation of the circle whose diameter is the common chord of the circles $x^2+y^2+2x+3y+2=0$ and $x^2+y^2+2x-3y-4=0$ 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:

Chords of the curve $4x^2 + y^2 - x + 4y = 0$ which subtend a right angle at the origin pass through a fixed point whose coordinates are:

The equation of the chord of contact of the circle $x^2 + y^2 + 4x + 6y - 12 = 0$ with respect to the point $(2, 3)$ is:

The circles $x^2 + y^2 = 9$ and $x^2 + y^2 - 12y + 27 = 0$ touch each other. The equation of their common tangent is