Question

Find the equation of the Parabola whose focus is $(-8,-2)$ and directrix is $y=2x-9$.

Answer 1

The distance of any point $P(h,k)$ on a parabola from the focus is equal to its perpendicular distance from the directrix.

$\therefore \sqrt{(h+8{)}^2+(k+2{)}^2}=\left|\frac{k-2h+9}{\sqrt{(-2{)}^2+1^2}}\right|$

Squaring on both sides,

$(h+8{)}^2+(k+2{)}^2=\frac{(k-2h+9{)}^2}{5}$

$\therefore 5(h^2+16h+64+k^2+4k+4)=k^2+4h^2+81-4hk-36h+18k$

$\therefore 5h^2+80h+320+5k^2+20k+20=k^2+4h^2+81-4hk-36h+18k$

$\therefore h^2+4k^2+4hk+116h+2k+259=0$

Hence, the required equation of parabola is $x^2+4y^2+4xy+116x+2y+259=0$