Q. The equation of 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)
Given focus for parabola is $S$(0,0) and equation of directrix is $x + y = 4$
Let $P(x,y)\$be any point on the parabola
Then, $SP^{2} = PM^{2}$
$(x - 0)^{2} + (y - 0)^{2} = \left\lbrack \frac{x + y - 4}{\sqrt{1 + 1}} \right\rbrack^{2}$
$\Rightarrow \ x^{2} + y^{2} = \frac{x^{2} + y^{2} + 16 + 2xy - 8y - 8x}{2}$
$\Rightarrow \ \ x^{2} + y^{2} - 2xy + 8x + 8y - 16 = 0$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.