Let $A=(2,3)$ and $B=(3,-5)$ be two vertices of $\triangle ABC$ such that $C$ is a point on the line $L \equiv 3x + 4y - 5 = 0$. Then the locus of the centroid of $\triangle ABC$ is a line parallel to

For the variable $t$, the locus of the point of intersection of the lines $3tx - 2y + 6t = 0$ and $3x + 2ty - 6 = 0$ is

If $P(x, y)$ is a variable point which is at a distance of $2$ units from the line $2x - 3y + 1 = 0$ and $\sqrt{13}$ units from the point $(5, 6)$,then the equation of the locus of $P$ is:

Two points $A(-a, 0)$ and $B(a, 0)$ are given. If $C$ is a variable point lying on one side of the line $AB$ such that $\angle CAB - \angle CBA = \alpha$,where $\alpha$ is a positive constant,then the locus of the point $C$ is

If a variable line drawn through the point of intersection of straight lines $\frac{x}{\alpha} + \frac{y}{\beta} = 1$ and $\frac{x}{\beta} + \frac{y}{\alpha} = 1$ meets the coordinate axes in $A$ and $B$,then the locus of the midpoint of $AB$ is

The ends of a rod of length $l$ move on two mutually perpendicular lines. The locus of the point on the rod which divides it in the ratio $1 : 2$ is