The image of the point $(4, -3)$ with respect to the line $y = x$ is

Let the mirror image of point $A(1, 4)$ in the line $y = x$ be point $B$; the mirror image of point $B$ in the line $y = -x$ be $C$; and the mirror image of $C$ in the $x$-axis be $D$. Then,the area of triangle $ABD$ is ............... $sq. \, units$.

If $P^{\prime}(a, b)$ is the image of the point $P(-1, 2)$ with respect to the line $x-2y+3=0$,then the length of the perpendicular from $P^{\prime}$ onto the line $2x+y-7=0$ is

If $(\alpha, \beta)$ is the image of the point $(3, -4)$ with respect to the line $4x - y - 1 = 0$,then the value of $\beta - \alpha$ is equal to

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 $k=$

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