If $B$ is the reflection of the point $A(1, 2)$ with respect to the line $y = x$ and $(\alpha, \beta)$ is the reflection of $B$ with respect to $y = 0$,then-

If $PM$ is the perpendicular from $P(2, 3)$ onto the line $x + y = 3$,then the coordinates of $M$ are

Let $ABC$ be a triangle and $A=(1,2)$. If $x-3y-5=0$ and $x+5y-9=0$ are the perpendicular bisectors of the sides $AB$ and $BC$ respectively,then the length of the side $AC$ 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:

Let $ABC$ be a triangle formed by the lines $7x-6y+3=0$,$x+2y-31=0$,and $9x-2y-19=0$. Let the point $(h, k)$ be the image of the centroid of $\Delta ABC$ in the line $3x+6y-53=0$. Then $h^2+k^2+hk$ is equal to

From the point $(-1, -1)$, two rays are sent making angles of $45^\circ$ with the line $x+y=0$. These rays get reflected from the mirror $x+2y=1$. If the equations of the reflected rays are $ax+by=9$ and $cx+dy=7$, where $a, b, c, d \in Z$, then the value of $ad+bc$ is . . . . . . .