Q. The equation of the parabola with focus $(0,0)$ and directrix $x + y = 4$, is
A) $x^{2} + y^{2} - 2xy + 8x + 8y - 16 = 0$
B) $x^{2} + y^{2} - 2xy + 8x + 8y = 0$
C) $x^{2} + y^{2} + 8x + 8y - 16 = 0$
D) $x^{2} - y^{2} + 8x + 8y - 16 = 0$
Explanation
(a)
We know, $SP = PM \Rightarrow {SP}^{2} = {PM}^{2}$
$\therefore\ {(x - 0)}^{2} + {(y - 0)}^{2} = \left( \frac{x + y - 4}{\sqrt{1^{2} + 1^{2}}} \right)^{2}$
$\Rightarrow x^{2} + y^{2} = \left( \frac{x + y - 4}{\sqrt{2}} \right)^{2}$
$\Rightarrow 2x^{2} + 2y^{2} = x^{2} + y^{2} + 16 + 2xy - 8y - 8x$
$\Rightarrow x^{2} + y^{2} - 2xy + 8x + 8y - 16 = 0$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.