$A$ point $P(x, y)$ is such that the sum of the squares of its distances from $(a, 0)$ and $(-a, 0)$ is $2b^2$. The equation representing the locus of $P$ 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

The locus of the midpoints of the chords of the circle $x^2 + y^2 = a^2$ which are parallel to $y = 2x$ 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 $(1, 0)$?

If the sum of the distances from a variable point $P$ to the given points $A(1,0)$ and $B(0,1)$ is $2$,then the locus of $P$ is

From the point $P(3, 4)$,chords are drawn to the circle $x^2 + y^2 - 4x = 0$. Find the locus of the midpoints of these chords.