If a circle $S$ passing through the point $(3,4)$ cuts the circle $x^2+y^2=36$ orthogonally,then the locus of the centre of $S$ is

Let $A$ be the point $(1, 2)$ and $B$ be any point on the curve $x^2 + y^2 = 16$. If the centre of the locus of the point $P$,which divides the line segment $AB$ in the ratio $3:2$ is the point $C(\alpha, \beta)$,then the length of the line segment $AC$ 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

The locus of the centre of a circle,which touches two given circles externally,is

The locus of the centers of the circles which cut the circles $x^2 + y^2 + 4x - 6y + 9 = 0$ and $x^2 + y^2 - 5x + 4y - 2 = 0$ orthogonally is

Two line segments $AB$ and $CD$ are constrained to move along the $X$ and $Y$-axes,respectively,in such a way that the points $A, B, C, D$ are concyclic. If $AB = a$ and $CD = b$,then the locus of the centre of the circle passing through $A, B, C, D$ in polar coordinates is