The image of $A(1, -2)$ with respect to the straight line $L \equiv 2x - 3y + 5 = 0$ is $B$. The foot of the perpendicular from $P(-4, -1)$ onto the line joining $AB$ is:

$A$ ray of light passing through the point $P(2, 3)$ reflects on the $x$-axis at point $A$ and the reflected ray passes through the point $Q(5, 4)$. Let $R$ be the point that divides the line segment $AQ$ internally in the ratio $2:1$. Let the coordinates of the foot of the perpendicular $M$ from $R$ on the bisector of the angle $PAQ$ be $(\alpha, \beta)$. Then,the value of $7\alpha + 3\beta$ is equal to.......

The distance from the origin to the image of $(1,1)$ with respect to the line $x+y+5=0$ is (in $\sqrt{2}$)

The equation of the line bisecting perpendicularly the segment joining the points $(-4, 6)$ and $(8, 8)$ is

$A$ ray of light coming from the point $(1, 2)$ is reflected at a point $A$ on the $x$-axis and then passes through the point $(5, 3)$. The coordinates of the point $A$ are:

The coordinates of the foot of the perpendicular from the point $(2, 3)$ on the line $x + y - 11 = 0$ are