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

The area of the triangle formed by the intersection of a line parallel to $X$-axis and passing through $P(h, k)$ with the lines $y=x$ and $x+y=2$ is $h^{2}$. The locus of the point $P$ is

If the sum of the perpendicular distances of a variable point $P(x, y)$ from the lines $x + y - 5 = 0$ and $3x - 2y + 7 = 0$ is always $10$,show that $P$ must move on a line.

The locus of points $(x, y)$ in the plane satisfying $\sin ^2 x + \sin ^2 y = 1$ consists of

$A$ point $P$ moves inside a square of area $4$ square units such that it is nearer to the point of intersection of its diagonals than any vertex. The area of the region traced by $P$ is

The line $\frac{x}{a} + \frac{y}{b} = 1$ moves in such a way that $\frac{1}{a^2} + \frac{1}{b^2} = \frac{1}{2c^2},$ where $a, b, c \in R_0$ and $c$ is a constant. Then,the locus of the foot of the perpendicular from the origin on the given line is -