Question

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

Answer 1

Answer: 8

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