Find the equation of a parabola with focal point $(-8, -2)$ and directrix $y -2 x + 9 = 0$

Question

Find the equation of a parabola with focal point $(-8, -2)$ and directrix $y -2 x + 9 = 0$

The equation I got was : $(y+3)^2=-17(x+2)$ but it seems to be wrong. Please help.

Answer 1

By the defintion of parabola, distance of point $P(x,y)$ from focus and directrix are equal

$$\Rightarrow \cfrac{|y-2x+9|}{\sqrt{5}} = \sqrt{(x+8)^2 + (y+2)^2} $$ On squaring and multiplying both sides by $5$, $$ y^2 + 4x^2 + 81 - 4xy - 36x + 18y = 5x^2 + 320 + 80x + 5y^2 + 20 + 20y$$ $$\Rightarrow x^2 + 4y^2 + 259 + 4xy + 116x + 2y =0$$

Answer 2

If the focus is $(f_x,f_y)$ and the directrix is $ax+by+c=0$, then the equation of the parabola is,

$$\frac{(ax+by+c)^2}{a^2+b^2} = (x-f_x)^2+(y-f_y)^2 $$

So,

$$\frac{(y-2x+9)^2}{1+4} = (x+8)^2+(y+2)^2 $$

$$(y-2x+9)^2 = 5\big[(x+8)^2+(y+2)^2\big] $$