question_answer

Let S and S' be two foci of the ellipse\[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\]if a circle described on SS' as diameter intersects the ellipse at real and distinct points, then the eccentricity e of the ellipse satisfies

A) \[e=1/\sqrt{2}\]         

B) \[e\in (1/\sqrt{2,}1)\]

C) \[e\in (0,1/\sqrt{2,\,})\]

D) None of these

Correct Answer: B

Solution :

[b] The radius of circle having SS' as diameter is r=ae. If it cuts an ellipse, then r>b or  ae>b or \[{{e}^{2}}>\frac{{{b}^{2}}}{{{a}^{2}}}\] or \[{{e}^{2}}>1-{{e}^{2}}\] or \[{{e}^{2}}>\frac{1}{2}\] or \[e>\frac{1}{\sqrt{2}}\] or \[e\in \left( \frac{1}{\sqrt{2}},1 \right)\]