$A$ 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.

Consider the circle $C : x^2 + y^2 - 6x - 8y - 11 = 0$. Let a variable chord $AB$ of the circle $C$ subtend a right angle at the origin. If the locus of the foot of the perpendicular drawn from the origin on the chord $AB$ is the circle $x^2 + y^2 - \alpha x - \beta y - \gamma = 0$,then $\alpha + \beta + 2\gamma$ is equal to . . . . . .

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

The locus of the point which is equidistant from the point $(1,1)$ and the line $x+y+1=0$ is

The angle between the pair of tangents from a point $P$ to the circle $x^2 + y^2 + 4x - 6y + 9\sin^2\alpha + 13\cos^2\alpha = 0$ is $2\alpha$. The equation of the locus of $P$ is...

What is the equation of the locus of a point which moves such that $4$ times its distance from the $x$-axis is the square of its distance from the origin?