$A$ 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.

If a system of circles passes through $(2,3)$ and cuts the circle $x^2+y^2=12$ orthogonally,then the equation of the locus of the centres of that system of circles is:

$A$ 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:

From a point $A(1, 0)$ on the circle $x^2+y^2-2x+2y+1=0$,a chord $AB$ is drawn and it is extended to a point $P$ such that $AP=3AB$. The equation of the locus of $P$ is

If $P$ is a point such that the ratio of the square of the lengths of the tangents from $P$ to the circles $x^2+y^2+2x-4y-20=0$ and $x^2+y^2-4x+2y-44=0$ is $2:3$,then the locus of $P$ is a circle with centre :

The line segment joining the points $(-5, 1)$ and $(3, 2)$ subtends a right angle at a variable point $P$. Find the locus of the point $P$.