The value of $a$ for which the image of the point $(a, a - 1)$ with respect to the line mirror $3x + y = 6a$ is the point $(a^2 + 1, a)$ 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:

$A$ ray of light passing through the point $(2, 3)$ reflects on the $Y$-axis at a point $P$. If the reflected ray passes through the point $(3, 2)$ and $P = (a, b)$,then $5b =$

The image of the point $(4, -13)$ with respect to the line $5x + y + 6 = 0$ is

Let $L$ be the line $y = 2x$ in the two-dimensional plane.
Statement $1$: The image of the point $(0, 1)$ in $L$ is the point $\left( \frac{4}{5}, \frac{3}{5} \right)$.
Statement $2$: The points $(0, 1)$ and $\left( \frac{4}{5}, \frac{3}{5} \right)$ lie on opposite sides of the line $L$ and are at equal distance from it.

The perpendicular from the origin to a line meets it at the point $(-2, 9)$. Find the equation of the line.