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:

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...

Let $T'$ be the line passing through the points $P(-2, 7)$ and $Q(2, -5)$. Let $F_1$ be the set of all pairs of circles $(S_1, S_2)$ such that $T'$ is tangent to $S_1$ at $P$ and tangent to $S_2$ at $Q$,and also such that $S_1$ and $S_2$ touch each other at a point,say,$M$. Let $E_1$ be the set representing the locus of $M$ as the pair $(S_1, S_2)$ varies in $F_1$. Let the set of all straight line segments joining a pair of distinct points of $E_1$ and passing through the point $R(1, 1)$ be $F_2$. Let $E_2$ be the set of the mid-points of the line segments in the set $F_2$. Then,which of the following statement$(s)$ is (are) $TRUE$?

Let $A = (a, 0)$ and $B = (-a, 0)$ be two fixed points. For $a \in (-\infty, 0)$ and a point $P$ moving in the plane such that $PA = nPB$ $(n \neq 0)$. If $|n| \neq 1$,then the locus of point $P$ is....

$A$ rod of length $12 \, cm$ moves with its ends always touching the coordinate axes. Determine the equation of the locus of a point $P$ on the rod,which is $3 \, cm$ from the end in contact with the $x-$axis.

Without changing the direction of the coordinate axes,the origin is shifted to $(h, k)$ such that the linear (first-degree) terms in the equation $x^2 + y^2 - 4x + 6y - 7 = 0$ are eliminated. Then the point $(h, k)$ is: