Q. The parabola with directrix $x + 2y - 1 = 0$ and focus (1,0) is
A) $4x^{2} - 4xy + y^{2} - 8x + 4y + 4 = 0$
B) $4x^{2} + 4xy + y^{2} - 8x + 4y + 4 = 0$
C) $4x^{2} + 5xy + y^{2} + 8x - 4y + 4 = 0$
D) $4x^{2} - 4xy + y^{2} - 8x - 4y + 4 = 0$
Explanation
(a)
Let $P(x,y)$ be any point on the parabola. By definition of parabola $PM = PS$
$\frac{x + 2y - 1}{\sqrt{1 + 4}} = \sqrt{(x - 1)^{2} + y^{2}}$
$\Rightarrow \ x^{2} + 4y^{2} + 1 + 4xy - 4y - 2x = 5\left( x^{2} + 1 - 2x + y^{2} \right)$
$\Rightarrow \ 4x^{2} + y^{2} - 8x + 4y - 4xy + 4 = 0$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.