$A$ point moves in such a way that its distance from $(1, -2)$ is always twice its distance from $(-3, 5)$. The locus of the point is:

The locus of the mid-points of the chords of the circle $x^2+y^2=16$ which are tangents to the hyperbola $9x^2-16y^2=144$ is

$A$ point moves such that its distance from the point $(-1, 0)$ is always three times its distance from the point $(0, 2)$. The locus of the point is

Let $PQ$ and $MN$ be two straight lines touching the circle $x^{2}+y^{2}-4x-6y-3=0$ at the points $A$ and $B$ respectively. Let $O$ be the centre of the circle and $\angle AOB=\pi/3$. Then the locus of the point of intersection of the lines $PQ$ and $MN$ is:

The locus of the centre of circles passing through $(a, b)$ and cutting the circle $x^2+y^2-2x+4y-4=0$ orthogonally is

$A$ circle passes through the point $(3, 4)$ and cuts the circle $x^2 + y^2 = a^2$ orthogonally. The locus of its centre is a straight line. If the distance of this straight line from the origin is $25$,then $a^2$ is equal to: