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

The locus of the centre of a circle which passes through the point $(a, 0)$ and touches the line $x + 1 = 0$ is

$P$ is a variable point such that the distance of $P$ from $A(4,0)$ is twice the distance of $P$ from $B(-4,0)$. If the line $3y - 3x - 20 = 0$ intersects the locus of $P$ at the points $C$ and $D$,then the distance between $C$ and $D$ is:

What is the locus of the centroid of the triangle whose vertices are $(a \cos t, a \sin t)$,$(b \sin t, -b \cos t)$,and $(1, 0)$?

From the origin,chords are drawn to the circle $(x - 1)^2 + y^2 = 1$. The equation of the locus of the midpoints of these chords 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: