Let $A = (a, 0)$ and $B = (-a, 0)$ be two fixed points. For $a \in (-\infty, 0)$ and a point $P$ moving in the plane such that $PA = nPB$ $(n \neq 0)$. If $|n| \neq 1$,then the locus of point $P$ is....

Let $PQ$ and $MN$ be two straight lines touching the circle $x^{2}+y^{2}-4x-6y-3=0$ at the points $A$ and $B$ respectively. Let $O$ be the centre of the circle and $\angle AOB=\pi/3$. Then the locus of the point of intersection of the lines $PQ$ and $MN$ is:

The locus of the point of intersection of perpendicular tangents to the circle $x^{2}+y^{2}=16$ is

$A$ point moves such that the sum of the squares of its distances from the points $(1, 2)$ and $(-2, 1)$ is always $6$. Then, its locus is

If $A (c, 0)$ and $B (-c, 0)$ are two points,find the locus of point $P$ such that $PA^{2} + PB^{2} = AB^{2}$.

$A$ 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.