Q. Equation of the ellipse with eccentricity $\frac{1}{2}$ and foci at $( \pm 1,\ 0)$ is
A) $\frac{x^{2}}{3} + \frac{y^{2}}{4} = 1$
B) $\frac{x^{2}}{4} + \frac{y^{2}}{3} = 1$
C) $\frac{x^{2}}{4} + \frac{y^{2}}{3} = \frac{4}{3}$
D) None of these
Explanation
(b)
Given, $e = \frac{1}{2}$ and foci is $( \pm 1,\ 0)$
$\Rightarrow ae = 1 \Rightarrow a = \frac{1}{\frac{1}{2}} = 2$
Now, $b^{2} = a^{2}\left( 1 - e^{2} \right) = 2^{2}\left( 1 - \frac{1}{4} \right) = 4\left( \frac{3}{4} \right) = 3$
$\therefore$ The equation of ellipse is
$\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 \Rightarrow \ \frac{x^{2}}{4} + \frac{y^{2}}{3} = 1$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.