The locus of the point $P(x, y)$ such that the area of the $\triangle PAB$ is $7$,where $A(4, 5)$ and $B(-2, 3)$ are given points,is

The locus of the mid-points of the perpendiculars drawn from points on the line $x=2y$ to the line $x=y$ is:

The ends of a rod of length $l$ move on two mutually perpendicular lines. The locus of the point on the rod which divides it in the ratio $1 : 2$ is

$A$ point starts moving from $(1, 2)$ and its projections on $x$ and $y$-axes are moving with velocities of $3 \ m/s$ and $2 \ m/s$ respectively. Its locus is

Two particles $A$ and $B$ move from rest along a straight line with constant accelerations $f$ and $h,$ respectively. If $A$ takes $m$ seconds more than $B$ and describes $n$ units more than that of $B$ to acquire the same speed, then

The point $P$ is equidistant from $A(1, 3)$,$B(-3, 5)$,and $C(5, -1)$. Then $PA$ is equal to: