Question

The equation of perpendicular bisectors of sides $AB,BC$ of $\Delta ABC$ are $x-y-5=0$ and $x+2y=0$ respectively. If $A\equiv (1,-2),C\equiv (\alpha ,\beta )$, then $\alpha +\beta$ is equal to

Answer 1

$B(h,k)$ is the image of $A$ with respect to $x-y-5=0$
So, $\frac{h-1}{1}=\frac{k+2}{-1}=-2\left(\frac{1+2-5}{2}\right)$
$\Rightarrow \frac{h-1}{1}=\frac{k+2}{-1}=2$
$\Rightarrow (h,k)=(3,-4)$

Now, $C$ is the image of $B$ with respect to $x+2y=0$
So, $\frac{\alpha -3}{1}=\frac{\beta +4}{2}=-2\left(\frac{3-8}{1+4}\right)$
$\Rightarrow \alpha =5,\beta =0\Rightarrow \alpha +\beta =5$