The foot of the perpendicular from the point $(2,4)$ upon the line $x+y=4$ 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:

The locus of the image of a variable point $P(\alpha, 2 \alpha-1)$ with respect to the line $3 x-2 y+4=0$ 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.......

If a particle moving along $x-2y-3=0$ gets reflected in a perpendicular direction upon hitting the line $3x-2y-5=0$,then the line of the movement of the particle after reflection is

The image of the family of lines $(\lambda + 2)x + (\lambda - 1)y - (8\lambda + 1) = 0$ in the line mirror $y = x$ is (where $\lambda$ and $\mu$ are parameters):