Question

Find the equation of parabola, whose focus is $(-3,0)$ and directrix is $x+5=0$.

Answer 1

Let $P(h,ft)$ be any variable at parabola.
According to definition of parabola,
$SP=PM$ or $S{P}^2=P{M}^2$
$\Rightarrow (h+3{)}^2+(k-0{)}^2={\left(\frac{h+5}{\sqrt{1^2}}\right)}^2$
$\Rightarrow h^2+6h+9+k^2=\frac{h^2+25+10h}{1}$
$\Rightarrow h^2+6h+k^2+9-h^2-10h-25=0$
$\Rightarrow k^2-4h-16=0$
$\Rightarrow k^2=4h+16$
Thus, locus of point $P(h,k)$ is $y^2=4x+16$, which is required.