Question

The equation of the parabola with focus (0, 0) and directrix x + y = 4 is
(a) x² + y² − 2xy + 8x + 8y − 16 = 0
(b) x² + y² − 2xy + 8x + 8y = 0
(c) x² + y² + 8x + 8y − 16 = 0
(d) x² − y² + 8x + 8y − 16 = 0

Answer 1

(a) x² + y² − 2xy + 8x + 8y − 16 = 0

Let P (x, y) be any point on the parabola whose focus is S (0, 0) and the directrix is x + y = 4.

Draw PM perpendicular to x + y = 4.
Then, we have:
$$\begin{gathered} SP=PM \\ \Rightarrow S{P}^2=P{M}^2 \\ \Rightarrow {\left(x-0\right)}^2+{\left(y-0\right)}^2={\left(\frac{x+y-4}{\sqrt{2}}\right)}^2 \\ \Rightarrow x^2+y^2={\left(\frac{x+y-4}{\sqrt{2}}\right)}^2 \\ \Rightarrow 2x^2+2y^2=x^2+y^2+16+2xy-8x-8y \\ \Rightarrow x^2+y^2-2xy+8x+8y-16=0 \end{gathered}$$