Question

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

Answer 1

The equation of the parabola is:

$$4x^2 + 4xy + y^2 + 4x + 32y + 16 = 0$$

Answer 2

A point P(x,y) on the parabola is equidistant from the focus S(-1,-2) and the directrix $x-2y+3=0$. The distance from P to S is $PS = \sqrt{(x+1)^2 + (y+2)^2}$. The distance from P to the directrix is $PM = \frac{|x-2y+3|}{\sqrt{1^2+(-2)^2}} = \frac{|x-2y+3|}{\sqrt{5}}$. Equating $PS^2 = PM^2$: $5((x+1)^2 + (y+2)^2) = (x-2y+3)^2$ $5(x^2+2x+1 + y^2+4y+4) = x^2+4y^2+9-4xy-12y+6x$ $5x^2+10x+5 + 5y^2+20y+20 = x^2+4y^2+9-4xy-12y+6x$ Rearranging terms: $4x^2 + 4xy + y^2 + 4x + 32y + 16 = 0$