If $Q$ and $R$ are the images of the point $P(2,3)$ with respect to the lines $x-y+2=0$ and $2x+y-2=0$ respectively,then $Q$ and $R$ lie on

$A$ line perpendicular to the line segment joining the points $(1,0)$ and $(2,3)$ divides it in the ratio $1:n$. Find the equation of the line.

Sum of values of $k$ for which the image of the point $(\lambda^2 + 1, \lambda)$ with respect to the line $y = -3x + 6k$ is the point $(\lambda, \lambda - 1)$ is:

Let $A \left(\frac{3}{\sqrt{a}}, \sqrt{a}\right)$ with $a > 0$ be a fixed point in the $xy$-plane. The image of $A$ in the $y$-axis is $B$,and the image of $B$ in the $x$-axis is $C$. If $D(3 \cos \theta, a \sin \theta)$ is a point in the fourth quadrant such that the maximum area of $\triangle ACD$ is $12$ square units,then $a$ is equal to:

The equation of the line bisecting the line segment joining the points $(a, b)$ and $(a', b')$ at a right angle is

The image of the point $(3,8)$ in the line $x+3y=7$ is