Question

If the mirror image of point $(1,2)$ in the line $2y=x$ is $(a,b)$, then the value of $5(a+b)$ is:

Answer 1

Let $A\equiv (1,2)$ and $B\equiv (a,b)$
Midpoint of $(1,2)$ and $(a,b)$ will be $(\frac{1+a}{2},\frac{2+b}{2})$ which lies on $2y=x$
$\therefore \frac{1+a}{2}=2+b$
$\Rightarrow a=2b+3\cdots \left(1\right)$
Since, line $AB$ and line $2y=x$ are perpendicular.
$\Rightarrow$ Slope of line $AB\times \frac{1}{2}=-1$
$\Rightarrow$ Slope of line $AB=-2$
$\Rightarrow \frac{2-b}{1-a}=-2$
$\Rightarrow 2a=4-b\cdots \left(2\right)$
Solving equations $\left(1\right)$ and $\left(2\right)$
We get $a=\frac{11}{5}$ and $b=\frac{-2}{5}$
$\Rightarrow 5(a+b)=9$

Alternate Solution:
Using the image formula
$\frac{a-1}{1}=\frac{b-2}{-2}=-\frac{2(1-2\cdot 2)}{1^2+2^2}\Rightarrow a=11/5;y=-2/5$
So $5(a+b)=9$