The equation of the director circle of the circle ${x^2} + {y^2} = {a^2}$ is

If a circle passing through $A(1,1)$ touches the $X$-axis,then the locus of the other end of the diameter through $A$ is

$A$ variable circle passes through a fixed point $A(p, q)$ and touches the $x$-axis. The locus of the other end of the diameter passing through $A$ is:

The equation $\sqrt{(x - 2)^2 + y^2} + \sqrt{(x + 2)^2 + y^2} = 4$ represents a

The locus of the centroid of the triangle with vertices at $(a \cos \theta, a \sin \theta)$,$(b \sin \theta, -b \cos \theta)$ and $(1, 0)$ is (where $\theta$ is a parameter).

The locus of the mid-points of the chords of the circle $x^2 + y^2 - 2x - 4y - 11 = 0$ which subtend $60^o$ at the centre is