Question

Let $L_1: x-y-2=0$ and $L_2: 2 x+y+1=0$ are the tangents to a parabola, whose focus is $(1,2)$.
If equation of its directrix is $a x+b y+3=0$, then find the value of $(a+b)$.

Answer 1

Image of focus $(1,2)$ in the tangent $L_1: x-y-2=0$
$\frac{x-1}{1}=\frac{y-2}{-1}=-2\left(\frac{1-2-2}{2}\right) $
$x=4, y=-1 \Rightarrow A \equiv(4,-1)$
Image of focus $(1,2)$ in the tangent $L_2: 2 x+y+1=0$
$\frac{x-1}{2}=\frac{y-2}{1}=-2\left(\frac{2+2+1}{5}\right) $
$x=-3, y=0 \Rightarrow B \equiv(-3,0)$
$\Theta$ A\& $B$ lie on the directrix, therefore equation of directrix is (line AB)
$ y-0=\frac{-1}{7}(x+3) $
$\Rightarrow x+7 y+3=0 \equiv a x+b y+3=0 $
$\Rightarrow a+b=8$