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) \[9{{x}^{2}}+5{{y}^{2}}=180\]
Correct Answer: D
Solution :
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 \[\frac{{{x}^{2}}}{20}+\frac{{{y}^{2}}}{36}=1\] \[\Rightarrow \] \[9{{x}^{2}}+5{{y}^{2}}=180\]
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