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

$A(a, 0)$ is a fixed point and $\theta$ is a parameter such that $0 < \theta < 2 \pi$. If $P(a \cos \theta, a \sin \theta)$ is a point on the circle $x^2+y^2=a^2$ and $Q(b \sin \theta, -b \cos \theta)$ is a point on the circle $x^2+y^2=b^2$,then the locus of the centroid of the triangle $APQ$ is

If the tangents to the circle $x^2 + y^2 = a^2$ with slopes $\alpha$ and $\beta$ intersect at point $P$,and $\cot \alpha + \cot \beta = 0$,then the locus of $P$ is:

Let $A=(2,0)$ and $B=(0,-2)$. Let $P$ be any point such that the sum of the distances of $P$ from $A$ and $B$ is $4$. Then the equation of the locus of the point $P$ is

Any circle passing through the point of intersection of the lines $x + \sqrt{3}y = 1$ and $\sqrt{3}x - y = 2$ intersects these lines at points $P$ and $Q$. The angle subtended by the arc $PQ$ at its centre is- ............. $^o$

Let $A$ be the point $(3, 0)$ and circles with variable diameter $AB$ touch the circle $x^2 + y^2 = 36$ internally. Let the curve $C$ be the locus of the point $B$. If the eccentricity of $C$ is $e$, then $72e^2$ is equal to . . . . . .