A) \[\frac{{{x}^{2}}}{12}-\frac{{{y}^{2}}}{4}=1\]
B) \[\frac{{{x}^{2}}}{4}-\frac{{{y}^{2}}}{12}=1\]
C) \[\frac{{{x}^{2}}}{8}-\frac{{{y}^{2}}}{2}=1\]
D) \[\frac{{{x}^{2}}}{16}-\frac{{{y}^{2}}}{9}=1\]
Correct Answer: B
Solution :
Let the equation of hyperbola is \[\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1\] Given, \[e=2,\text{ }2ne=8\] \[\Rightarrow \] \[ae=4\Rightarrow a=2\] Now. \[{{b}^{2}}={{a}^{2}}\,({{e}^{2}}-1)\] \[\Rightarrow \] \[{{b}^{2}}=4\,(4-1)\,\,\Rightarrow \,\,{{b}^{2}}=12\] \[\therefore \] equation of hyperbola is \[\frac{{{x}^{2}}}{4}-\frac{{{y}^{2}}}{12}=1\]
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