Every point on the curve $3 x+2 y-3 x y=0$ is the centroid of a triangle formed by the coordinate axes and a line $(L)$ intersecting both the coordinates axes. Then, all such lines $(L)$

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 $\tan 2 \theta=$