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

$A$ man starts walking from the point $P(-3, 4)$,touches the $x$-axis at $R$,and then turns to reach the point $Q(0, 2)$. The man is walking at a constant speed. If the man reaches the point $Q$ in the minimum time,then $50((PR)^{2} + (RQ)^{2})$ is equal to ..... .

The locus of the midpoint of the portion of the line $x \cos \alpha + y \sin \alpha = p$ intercepted by the coordinate axes,where $p$ is a constant,is

Consider a square $ABCD$ of side $12$ and let $M, N$ be the midpoints of $AB, CD$ respectively. Take a point $P$ on $MN$ and let $AP=r, PC=s$. Then,the area of the triangle whose sides are $r, s, 12$ is

$p, x_1, x_2, \ldots, x_n$ and $q, y_1, y_2, \ldots, y_n$ are two arithmetic progressions with common differences $a$ and $b$ respectively. If $\alpha$ and $\beta$ are the arithmetic means of $x_1, x_2, \ldots, x_n$ and $y_1, y_2, \ldots, y_n$ respectively,then the locus of $P(\alpha, \beta)$ is

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