Let $A, B$ and $C$ be three points in a plane. The locus of a point $P$ moving such that $PA^2 + PB^2 = 2PC^2$ is a

$ABC$ is an isosceles triangle. If the coordinates of the base are $(1, 3)$ and $(-2, 7)$,then the coordinates of vertex $A$ can be:

If a variable line drawn through the intersection of the lines $\frac{x}{3} + \frac{y}{4} = 1$ and $\frac{x}{4} + \frac{y}{3} = 1$ meets the coordinate axes at $A$ and $B$ $(A \neq B)$,then the locus of the midpoint of $AB$ is

If $P_1, P_2, P_3, \ldots, P_n$ are $n$ points on the line $y=x$ all lying in the first quadrant,such that $OP_n = n(OP_{n-1})$ ($O$ is origin),$OP_1 = 1$ and $P_n = (2520 \sqrt{2}, 2520 \sqrt{2})$,then $n=$

If a variable straight line passing through the point of intersection of the lines $x-2y+3=0$ and $2x-y-1=0$ intersects the $X, Y$-axes at $A$ and $B$ respectively,then the equation of the locus of a point which divides the segment $AB$ in the ratio $-2:3$ is

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