The locus of the midpoints of the chords of the circle $x^2 + y^2 = a^2$ which are parallel to $y = 2x$ is:

The locus of the point of intersection of the lines,$\sqrt{2}x - y + 4\sqrt{2}k = 0$ and $\sqrt{2}kx + ky - 4\sqrt{2} = 0$ (where $k$ is any non-zero real parameter) is

The locus of the point whose ratio of distance from the origin to its distance from $(-2, -3)$ is $5: 7$,is given by:

The equation to the locus of a point which moves so that its distance from the $x$-axis is always one-half its distance from the origin 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 $(1, 0)$?

Let $ABCD$ and $AEFG$ be squares of side $4$ and $2$ units,respectively. The point $E$ is on the line segment $AB$ and the point $F$ is on the diagonal $AC$. Then the radius $r$ of the circle passing through the point $F$ and touching the line segments $BC$ and $CD$ satisfies: