If $P = (1, 0)$,$Q = (-1, 0)$,and $R = (2, 0)$ are three given points,then the locus of the points $S$ satisfying the relation $SQ^2 + SR^2 = 2 SP^2$ is :

If $A=(-1, 2)$ and $B=(1, -2)$ are two points and $P$ is a variable point such that the area of $\triangle PAB$ is always $1$,then the equation of the locus of $P$ is

$O$ is the origin and $A$ is the point $(3,4)$. If a point $P(x,y)$ moves such that the line segment $OP$ is always parallel to the line segment $OA$,then the equation of the locus of $P$ is:

$A$ line cuts the $x$-axis at $A(7, 0)$ and the $y$-axis at $B(0, -5)$. $A$ variable line $PQ$ is drawn perpendicular to $AB$ cutting the $x$-axis at $P(a, 0)$ and the $y$-axis at $Q(0, b)$. If $AQ$ and $BP$ intersect at $R(h, k)$, the locus of $R$ is

$A$ man standing on a railway platform noticed that a train took $21\,s$ to cross the platform (this means the time elapsed from the moment the engine enters the platform till the last compartment leaves the platform),which is $88\,m$ long,and that it took $9\,s$ to pass him. Assuming that the train was moving with uniform speed,what is the length of the train in meters?

Let $A\ (2, -3)$ and $B\ (-2, 1)$ be the vertices of a triangle $ABC$. If the centroid of this triangle moves on the line $2x + 3y = 1$,then the locus of the vertex $C$ is which line?