$A$ and $B$ are fixed points with coordinates $(0, a)$ and $(0, b)$ where $a > 0$ and $b > 0$. $P$ is a variable point $(x, 0)$ on the $x$-axis. If the angle $\angle APB$ is maximum,then:

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

$A$ chord $AB$ is drawn from the point $A(0,3)$ on the circle $x^2 + 4x + (y - 3)^2 = 0$. If the point $M$ lies on the chord such that $AM = 2AB$,then the locus of point $M$ is:

$A$ point $P$ moves in such a way that the ratio of its distance from two coplanar points is always a fixed number $(\lambda \ne 1)$. Then its locus 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 tangent at $P$,any point on the circle ${x^2} + {y^2} = 4$,meets the coordinate axes in $A$ and $B$,then