The reflection of the point $(4, -13)$ in the line $5x + y + 6 = 0$ 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 equation of the straight line passing through the point $(a \cos^3 \theta, a \sin^3 \theta)$ and perpendicular to the line $x \sec \theta + y \csc \theta = a$ is:

Let a ray of light passing through the point $(3,10)$ reflect on the line $2x+y=6$ and the reflected ray pass through the point $(7,2)$. If the equation of the incident ray is $ax+by+1=0$,then $a^2+b^2+3ab$ is equal to:

Let $\alpha$ and $\beta$ be integers satisfying $0 < \beta < \alpha$. Let $P(\alpha, \beta)$ be a point. Let $Q$ be the reflection of $P$ in the line $y = x$,$R$ be the reflection of $Q$ in the $y$-axis,$S$ be the reflection of $R$ in the $x$-axis,and $T$ be the reflection of $S$ in the $y$-axis. If the area of the convex pentagon $PQRST$ is $187 \ sq. \ units$,then the value of $\alpha + \beta^2$ is:

If the perpendicular bisector of the line segment joining $P(1, 4)$ and $Q(k, 3)$ has a $y$-intercept of $-4$,then the possible value of $k$ is: