A) \[3{{x}^{2}}+4{{y}^{2}}=1\]
B) \[3{{x}^{2}}+4{{y}^{2}}=12\]
C) \[4{{x}^{2}}+3{{y}^{2}}=12\]
D) \[4{{x}^{2}}+3{{y}^{2}}=1\]
Correct Answer: B
Solution :
The equation of directrix of an ellipse\[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\]is\[x=\pm \frac{a}{e}\]. Since, directrix is\[x=4,\]then major axis of an ellipse is along x-axis. \[\therefore \] \[\frac{a}{e}=4\Rightarrow a=4e\] \[\Rightarrow \]\[a=4\times \frac{1}{2}\]\[\Rightarrow \]\[a=2\] \[\left( \because \theta =\frac{1}{2} \right)\] Also, we know that \[{{b}^{2}}={{a}^{2}}(1-{{e}^{2}})\] \[{{b}^{2}}=4\left( 1-\frac{1}{4} \right)=4\times \frac{3}{4}\] \[\Rightarrow \] \[{{b}^{2}}=3\] \[\therefore \]Equation of ellipse is \[\frac{{{x}^{2}}}{4}+\frac{{{y}^{2}}}{3}=1\] \[\Rightarrow \]\[3{{x}^{2}}+4{{y}^{2}}=12\]
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