The sum of the squares of the distances of a moving point from two fixed points $A(a, 0)$ and $B(-a, 0)$ is equal to a constant $2c^2$. Then,the equation of its locus is:

The locus of the midpoint of a chord of the circle $x^{2} + y^{2} = 4$ which subtends an angle of $45^{\circ}$ at the major arc of the circle is:

$A$ rod of length $12 \, cm$ moves with its ends always touching the coordinate axes. Determine the equation of the locus of a point $P$ on the rod,which is $3 \, cm$ from the end in contact with the $x-$axis.

$A$ 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 locus of a point $P$ which divides the line joining $(1, 0)$ and $(2\cos \theta, 2\sin \theta)$ internally in the ratio $2 : 3$ for all $\theta$ is a

$A$ variable circle is drawn passing through the origin $O$. It intersects the $X$ and $Y$ axes at points $A$ and $B$ respectively,such that $OA + 2OB = K$ (a non-zero constant). The circle always passes through a fixed point $P$ other than the origin. The point $P$ lies on -