A) \[4\sqrt{2}\]
B) \[8\sqrt{2}\]
C) \[2\sqrt{2}\]
D) \[\sqrt{2}\]
Correct Answer: D
Solution :
Given equation of hyperbola is \[{{x}^{2}}-{{y}^{2}}=4\] or \[\frac{{{x}^{2}}}{4}-\frac{{{y}^{2}}}{4}=1\] \[(\because \,\,{{a}^{2}}={{b}^{2}}=4)\] Now, \[{{b}^{2}}={{a}^{2}}({{e}^{2}}-1)\] \[4=4({{e}^{2}}-1)\Rightarrow e=\pm \sqrt{2}\] Equation of directrix \[x=\pm \frac{a}{e}\] \[\Rightarrow \] \[x=\pm \frac{2}{\sqrt{2}}\] \[\Rightarrow \] \[x=\pm \sqrt{2}\Rightarrow x-\sqrt{2}=0\] and the focus \[=(\pm ae,0)\] \[=(\pm 2\sqrt{2},0)=(2\sqrt{2},0)\] (taken positive sign) The nearest distance of focus from directrix \[=\left| \frac{2\sqrt{2}-\sqrt{2}}{\sqrt{1}} \right|=\sqrt{2}\]
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