Two perpendicular tangents to the circle $x^2 + y^2 = a^2$ meet at point $P$. The equation of the locus of $P$ is:

Let the circle $x^{2}+y^{2}=4$ intersect the $x$-axis at the points $A(a,0), a>0$ and $B(b,0)$. Let $P(2 \cos \alpha, 2 \sin \alpha), 0 < \alpha < \frac{\pi}{2}$ and $Q(2 \cos \beta, 2 \sin \beta)$ be two points such that $(\alpha - \beta) = \frac{\pi}{2}$. Then the point of intersection of $AQ$ and $BP$ lies on:

$A$ circle cuts a chord of length $4a$ on the $x$-axis and passes through a point on the $y$-axis,distant $2b$ from the origin. Then the locus of the center of this circle is

$A$ variable straight line $AB$ divides the circumference of the circle $x^2 + y^2 = 25$ in the ratio $1 : 2$. If a tangent $CD$ is drawn to the smaller arc parallel to $AB$,such that $ABCD$ is a rectangle,then the locus of $C$ and $D$ is:

$A$ circle touches the $X$-axis and also touches the circle with radius $2$ and center at $(0, 3)$. Find the locus of the center of the circle.

The tangent at any point on the curve $x^2 + y^2 = r^2$ meets the coordinate axes at $A$ and $B$. If lines are drawn through $A$ and $B$ parallel to the coordinate axes to intersect at $P$,find the locus of $P$.