Question

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

Answer 1

The focus of the parabola is $F(-1,-2)$ and directrix is the line $x-2y+3=0$

Let $P(x,y)$ be any point in the plane of focus and directrix, and $MP$ be the perpendicular distance from $P$ to the directrix, then $P$ lies on the parabola iff $FP=MP$

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

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

$\Rightarrow \sqrt{{(x+1)}^2+{(y+2)}^2}=\frac{|x-2y+3|}{\sqrt{5}}$

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

$\Rightarrow 5[x^2+2x+1+y^2+4y+4]=x^2+4y^2+9-4xy-12y+6x$

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

$\Rightarrow 5x^2+10x+5y^2+20y+25-x^2-4y^2-9+4xy+12y-6x=0$

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