Question

The equation of the ellipse whose foci are $ (\pm ,2,0) $ and eccentricity $ \frac{1}{2} $ is:

• A. $ \frac{{{x}^{2}}}{12}+\frac{{{y}^{2}}}{16}=1 $
• B. $ \frac{{{x}^{2}}}{16}+\frac{{{y}^{2}}}{12}=1 $
• C. $ \frac{{{x}^{2}}}{16}+\frac{{{y}^{2}}}{8}=1 $
• D. none of these

Answer 1

Answer: (B)

Let the ellipse be $ \frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1 $ It is given that $ e=\frac{1}{2} $ and $ ae=2 $ Therefore, $ a=4 $ Now, $ {{b}^{2}}={{a}^{2}}(1-{{e}^{2}}) $ $ \Rightarrow $ $ {{b}^{2}}=12 $ Thus the required ellipse is $ \frac{{{x}^{2}}}{16}+\frac{{{y}^{2}}}{12}=1. $