The equation $x^2 + y^2 + 4x + 6y + 13 = 0$ represents

If a circle of a constant radius $6$ passes through the origin $O$ and meets the coordinate axes at $A$ and $B$, then find the locus of the centroid of $\triangle OAB$.

The locus of the centre of a circle which cuts orthogonally the circle $x^2 + y^2 - 20x + 4 = 0$ and which touches the line $x = 2$ is:

The locus of the moving point $P$,such that $2PA = 3PB$ where $A$ is $(0,0)$ and $B$ is $(4,-3)$,is

$A$ rod of length $6$ units slides with its ends on the coordinate axes. The locus of the midpoint of the rod is

For all values of $\theta$,the locus of the point of intersection of the lines $x \cos \theta + y \sin \theta = a$ and $x \sin \theta - y \cos \theta = b$ is