If $2x + 3y + 4 = 0$ is the perpendicular bisector of the line segment joining the points $A(1, 2)$ and $B(\alpha, \beta)$,then the value of $13\alpha + 13\beta$ equals $......$

Let the triangle $PQR$ be the image of the triangle with vertices $(1,3), (3,1)$ and $(2,4)$ in the line $x+2y=2$. If the centroid of $\triangle PQR$ is the point $(\alpha, \beta)$,then $15(\alpha-\beta)$ is equal to:

Find the equation of the right bisector of the line segment joining the points $(3, 4)$ and $(-1, 2).$

$A$ light ray along the line $x + \sqrt{3}y = \sqrt{3}$ reaches the $x$-axis and gets reflected. Find the equation of the reflected ray.

Let $P$ be a point inside a $\triangle ABC$ with $\angle ABC = 90^{\circ}$. Let $P_1$ and $P_2$ be the images of $P$ under reflection in $AB$ and $BC$ respectively. The distance between the circumcenters of $\triangle ABC$ and $\triangle P_1PP_2$ is

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