Question

Find the equation of the parabola whose focus is at (-1, -2) and the directrix is x - 2y + 3 = 0

Answer 1

Let $P(x,y)$ be any point on the parabola whose focus is $S(-1,-2)$ and the directrix $x-2y+3=0$.

Draw $PM$ perpendicular to directrix $x - 2y + 3 = 0$

Then by definition $SP=PM$

$\Rightarrow S{P}^2=P{M}^2$

$\Rightarrow (x+1{)}^2+(y+2{)}^2={\left(\frac{x-2y+3}{\sqrt{1+4}}\right)}^2$

$\Rightarrow 5\left[\right(x+1{)}^2+(y+2{)}^2]=(x-2y+3{)}^2$

$\Rightarrow 5(x^2+y^2+2x+4y+5)=(x^2+4y^2+9-4xy+6x-12y)$

$\Rightarrow 4x^2+y^2+4xy+4x+32y+16=0$ This is the equation of the required parabola