A) \[\sqrt{2}:1\]
B) \[\sqrt{3}:\sqrt{2}\]
C) \[1:2\]
D) \[2:1\]
Correct Answer: A
Solution :
Equation of hyperbola is \[\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1\] Distance between foci \[=2ae\] and distance between directrices\[=\frac{2a}{e}\] According to the question, \[\frac{2ae}{2a/e}=\frac{3}{2}\] \[\Rightarrow \] \[{{e}^{2}}=\frac{3}{2}\] We know, \[{{b}^{2}}={{a}^{2}}({{e}^{2}}-1)\] \[\Rightarrow \] \[\frac{{{b}^{2}}}{{{a}^{2}}}=\frac{3}{2}-1=\frac{1}{2}\] \[\Rightarrow \] \[\frac{b}{a}=\frac{1}{\sqrt{2}}\] \[\therefore \]Required ratio\[a:b\]is\[\sqrt{2}:1\].You need to login to perform this action.
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