Q. For an ellipse with eccentricity $1/2$ the centre is at the origin. If one directrix is $x = 4,$ then the equation of the ellipse is
A) $3x^{2} + 4y^{2} = 1$
B) $3x^{2} + 4y^{2} = 12$
C) $4x^{2} + 3y^{2} = 1$
D) $4x^{2} + 3y^{2} = 12$
Explanation
(b)
Given, $e = \frac{1}{2}$ and $\frac{a}{e} = 4\ \Rightarrow \ a = 2$
$\therefore\ \ b^{2} = a^{2}\left( 1 - e^{2} \right)$
$\Rightarrow \ b^{2} = 4\ \left( 1 - \frac{1}{4} \right) = \ 3\$
$\therefore$ Equation of ellipse is $\frac{x^{2}}{4} + \frac{y^{2}}{3} = 1\ \Rightarrow \ 3x^{2} + 4y^{2} = 12$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.