Let the triangle $PQR$ be the image of the triangle with vertices $(1,3), (3,1)$ and $(2,4)$ in the line $x+2y=2$. If the centroid of $\triangle PQR$ is the point $(\alpha, \beta)$,then $15(\alpha-\beta)$ is equal to:

The equation of a straight line which passes through the point $(a \cos^3 \theta, a \sin^3 \theta)$ and is perpendicular to $x \sec \theta + y \operatorname{cosec} \theta = a$ is

$A$ ray of light along $x + \sqrt{3}y = \sqrt{3}$ gets reflected upon reaching the $x$-axis. The equation of the reflected ray is:

Let $A \left(\frac{3}{\sqrt{a}}, \sqrt{a}\right)$ with $a > 0$ be a fixed point in the $xy$-plane. The image of $A$ in the $y$-axis is $B$,and the image of $B$ in the $x$-axis is $C$. If $D(3 \cos \theta, a \sin \theta)$ is a point in the fourth quadrant such that the maximum area of $\triangle ACD$ is $12$ square units,then $a$ is equal to:

Find the image of the point $(3,8)$ with respect to the line $x+3y=7$,assuming the line to be a plane mirror.

If the perpendicular drawn from the point $(2, -3)$ to the straight line $4x - 3y + 8 = 0$ meets it at $M(a, b)$ and $a^3 - b^3 = k^3$,then $k=$