The value of ' $a$ ' for which the equation $\left(a^2-3\right) x^2+16 x y -2 a y^2+4 x-8 y-2=0$ represents a pair of perpendicular lines is

If the points $A(2,3), B(3,2)$ form a triangle with a variable point $p\left(t, t^2\right)$, where $t$ is a parameter, then the equation of the locus of the centroid of $\triangle A B C$ is

If $(h, k)$ is the new origin to be chosen to eliminate first degree terms from the equation $S \equiv 2 x^2-x y-y^2-3 x+3 y=0$ by translation and if $\theta$ is the angle with which the axes are to be rotated about the origin in anti-clockwise direction to eliminate $x y$-term from $S=0$, then $\tan 2 \theta=$

A line $L$ perpendicular to the line $5 x-12 y+6=0$ makes positive intercept on the $Y$-axis. If the distance from the origin to the line $L$ is 2 units and the angle made by the perpendicular drawn from the origin to the line $L$ with positive $X$-axis is $\theta$, then $\tan \theta+\cot \theta=$