Question

The eccentricity of an ellipse, with its centre at the origin is $\frac{1}{2}$. If one of the directrices is $x = 4$, then the equation of the ellipse is

• A. $4x^{2} + 3y^{2} = 1$
• B. $3x^{2} + 4y^{2} = 12$
• C. $4x^{2} + 3y^{2} = 12$
• D. $3x^{2} + 4y^{2} = 1$

Answer 1

Answer: (B)

$3x^{2} + 4y^{2} = 12$

Answer 2

Given $e = \frac{1}{2},\frac{a}{e} = 4$. So, $a = 2 \Rightarrow a^{2} = 4$

From $b^{2} = a^{2}(1 - e^{2})$ ⇒ $b^{2} = 4\left( 1 - \frac{1}{4} \right) = 4 \times \frac{3}{4} = 3$

Hence the equation of ellipse is $\frac{x^{2}}{4} + \frac{y^{2}}{3} = 1$, i.e. $3x^{2} + 4y^{2} = 12$