Question

If the line, $2 x - y +3=0$ is at a distance $\frac{1}{\sqrt{5}}$ and $\frac{2}{\sqrt{5}}$ from the lines $4 x -2 y +\alpha=0$ and $6 x-3 y+\beta=0,$ respectively, then the sum of all possible values of $\alpha$ and $\beta$ is ______.

Answer 1

Apply distance between parallel line formula
$4 x-2 y+\alpha=0$
$4 x-2 y+6=0$
$\left|\frac{\alpha-6}{255}\right|=\frac{1}{55}$
$|\alpha-6|=2
\Rightarrow \alpha=8,4$
sum $=12$
again
$6 x-3 y+\beta=0$
$6 x-3 y+9=0$
$\left|\frac{\beta-9}{3 \sqrt{5}}\right|=\frac{2}{\sqrt{5}}$
$|\beta-9|=6 $
$\Rightarrow \beta=15,3$
sum$=18$
sum of all values of $\alpha$ and $\beta$ is $=30$