The locus of a point,such that the difference of the squares of the lengths of the tangents drawn from it to two given circles is constant,is:

Let $ABCD$ be a square. An arc of a circle with $A$ as center and $AB$ as radius is drawn inside the square joining the points $B$ and $D$. Points $P$ on $AB$,$S$ on $AD$,$Q$ and $R$ on $\operatorname{arc} BD$ are taken such that $PQRS$ is a square. Further suppose that $PQ$ and $RS$ are parallel to $AC$. Then,$\frac{\text{Area}(PQRS)}{\text{Area}(ABCD)}$ is

$A$ point $P(x, y)$ is such that the sum of the squares of its distances from $(a, 0)$ and $(-a, 0)$ is $2b^2$. The equation representing the locus of $P$ is

If a circle of constant radius $3k$ passes through the origin $O$ and meets the coordinate axes at $A$ and $B$,then the locus of the centroid of the triangle $OAB$ is

The locus of a point $(x, y)$ whose distance from the point $(-g, -f)$ is always $a$,where $k = g^2 + f^2 - a^2$,is:

If $A(2, 3)$ and $B(3, -2)$ are two fixed points and $P(x, y)$ is a variable point satisfying the condition $|PA - PB| = 2$,then the locus of $P$ is