Question

The foci of an ellipse are $ (0,\pm 4) $ and the equations for the directories are $ y=\pm 9 $ . The equation for the ellipse is

• A. $ 5{{x}^{2}}+9{{y}^{2}}=4 $
• B. $ 2{{x}^{2}}-6{{y}^{2}}=28 $
• C. $ 6{{x}^{2}}+3{{y}^{2}}=45 $
• D. $ 9{{x}^{2}}+5{{y}^{2}}=180 $

Answer 1

Answer: (D)

Given foci of ellipse are (0, - 4) and (0, 4). $ \therefore $ Focal distance is 8. $ \Rightarrow $ $ 2be=8 $ $ \Rightarrow $ $ be=4 $ ...(i) Also, since equation of directories are $ y=\pm 9 $ $ \Rightarrow $ $ \frac{b}{e}=9 $ ...(ii) $ \therefore $ From Eqs. (i) and (ii), we get $ {{b}^{2}}=36 $ $ \Rightarrow $ $ b=6 $ and $ e=\frac{2}{3} $ [from Eq. (i)] $ \therefore $ $ {{a}^{2}}={{b}^{2}}(1-{{e}^{2}})=36\left( 1-\frac{4}{9} \right)=\frac{36\times 5}{9}=20 $ So, equation of ellipse is $ \frac{{{x}^{2}}}{20}+\frac{{{y}^{2}}}{36}=1 $ $ \Rightarrow $ $ 9{{x}^{2}}+5{{y}^{2}}=180 $