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

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

If $m$ is the slope and $P(8, \beta)$ is the midpoint of a chord of contact of the circle $x^2+y^2=125$,then the number of values of $\beta$ such that $\beta$ and $m$ are integers is

If $A$ and $B$ are the points of contact of the tangents drawn from the point $P(-3, 1)$ to the circle $x^2+y^2-4x+2y-4=0$,then the equation of the circumcircle of the triangle $PAB$ is

Tangents are drawn from any point on the hyperbola $4x^2 - 9y^2 = 36$ to the circle $x^2 + y^2 = 9$. If the locus of the midpoint of the chord of contact is $\left( \frac{x^2}{9} - \frac{y^2}{4} \right) = \lambda \left( \frac{x^2 + y^2}{9} \right)^2$,then the value of $\lambda$ is:

The locus of the mid-point of the chord of contact of tangents drawn from points lying on the straight line $4x - 5y = 20$ to the circle $x^2 + y^2 = 9$ is