The locus of the midpoint of the chord of contact of tangents drawn from a point on the line $4x - 5y = 20$ to the circle $x^2 + y^2 = 9$ is:

For a real variable $a > 1$,consider the points $A_k = (k a, a^k)$,$k = 1, 2, \ldots, n$ in the Cartesian plane. If $\alpha$ and $\beta$ represent respectively the arithmetic mean of $x$-coordinates and the geometric mean of $y$-coordinates of $A_k$,then the locus of the point $P(\alpha, \beta)$ is

Two rods of lengths $a$ and $b$ slide along the coordinate axes in such a way that their ends are always concyclic. The locus of the center of the circle passing through the ends is:

$A$ point $P(x, y)$ divides the line segment joining the points $(5, 0)$ and $(10 \cos \theta, 10 \sin \theta)$ in the ratio $2 : 3$. Find the locus of point $P$ as $\theta$ varies.

$A$ point moves such that its distance from the point $(4, 0)$ is half that of its distance from the line $x = 16$. The locus of this point is

From the point $A(0, 3)$ on the circle $x^2 + 4x + (y - 3)^2 = 0$,a chord $AB$ is drawn and extended to a point $M$ such that $AM = 2 AB$. The equation of the locus of $M$ is: