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 :
Since directrix is\[x=4,\]then major axis of an ellipse is along\[x-\]axis. \[\therefore \] \[\frac{a}{e}=4\Rightarrow a=4e\Rightarrow 4\times \frac{1}{2}\] \[\Rightarrow \] \[a=2\] \[\left( \because e=\frac{1}{2} \right)\] Also we know that\[{{b}^{2}}={{a}^{2}}(1-{{e}^{2}})\] \[{{b}^{2}}=4\left( 1-\frac{1}{4} \right)=4\times \frac{3}{4}\] \[{{b}^{2}}=3\] \[\therefore \]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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