Equation of the ellipse with focus $(2, 0)$ , directrix $x = 8$ and $e = \frac{1}{2}$ is:

4 x^{2} + 3 y^{2} = 48

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3 x^{2} + 4 y^{2} = 48

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

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

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Solution

The correct option is D $3 x^{2} + 4 y^{2} = 48$
Distance between foci and directrix $=$

$a (\frac{1}{e} - e) = (8 - 2) = 6$

$a (\frac{3}{2}) = 6$

$a = 4$

$b^{2} = a^{2} (1 - e^{2}) = 16 (\frac{3}{4}) = 12$

$∴$ equation of ellipse is $\frac{x^{2}}{16} + \frac{y^{2}}{12} = 1$

$\Rightarrow 3 x^{2} + 4 y^{2} = 48$

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