Question

Find the equation of following parabola : focus is (1,1) and directrix is y + x + 2 = 0.

Answer 1

x^2 + y^2 - 2xy - 8x - 8y = 0

Answer 2

Let P(x,y) be any point on the parabola. By definition, the distance from P to the focus S(1,1) is equal to the perpendicular distance from P to the directrix $x+y+2=0$.

Distance PS = $\sqrt{(x-1)^2 + (y-1)^2}$ Distance PD = $\frac{|x+y+2|}{\sqrt{1^2+1^2}} = \frac{|x+y+2|}{\sqrt{2}}$

Equating the squares of the distances: $PS^2 = PD^2$ $(x-1)^2 + (y-1)^2 = \left(\frac{x+y+2}{\sqrt{2}}\right)^2$ $2[(x-1)^2 + (y-1)^2] = (x+y+2)^2$ $2[x^2 - 2x + 1 + y^2 - 2y + 1] = x^2 + y^2 + 4 + 2xy + 4x + 4y$ $2x^2 - 4x + 2 + 2y^2 - 4y + 2 = x^2 + y^2 + 2xy + 4x + 4y + 4$ $2x^2 + 2y^2 - 4x - 4y + 4 = x^2 + y^2 + 2xy + 4x + 4y + 4$

Rearranging the terms to one side: $(2x^2 - x^2) + (2y^2 - y^2) - 2xy + (-4x - 4x) + (-4y - 4y) + (4 - 4) = 0$ $x^2 + y^2 - 2xy - 8x - 8y = 0$

This can also be written as $(x-y)^2 - 8(x+y) = 0$.