question_answer

If the angle between the lines joining the end points of minor axis of an ellipse with its foci is \[{{x}^{2}}-{{y}^{2}}=25\], then the eccentricity of the ellipse is [IIT Screening 1997; Pb. CET 2001; DCE 2002]

A)            1/2 

B)            \[1/\sqrt{2}\]

C)            \[\sqrt{3}/2\]                            

D)            \[1/2\sqrt{2}\]

Correct Answer: B

Solution :

           Since \[\angle \,FBF'=\frac{\pi }{2}\] (Given) \[\therefore \]\[\angle \,FBC=\,\angle \,F'BC=\pi /4\]                      \[CB=CF\Rightarrow \,\,b=ae\] Þ \[{{b}^{2}}={{a}^{2}}{{e}^{2}}\]                   Þ \[{{a}^{2}}(1-{{e}^{2}})={{a}^{2}}{{e}^{2}}\]            Þ \[1-{{e}^{2}}={{e}^{2}}\] Þ \[2{{e}^{2}}=1\] Þ \[e=1/\sqrt{2}\].