Question

If the lines joining the foci of the ellipse $ \frac{{{x}^{2}}}{{{a}^{2}}}\,+\frac{{{y}^{2}}}{{{b}^{2}}}=1, $ where $ a>b, $ and an extremly of its minor axis are inclined at an angle $ 60{}^\circ , $ then the eccentricity of the ellipse is

• A. $ -\frac{\sqrt{3}}{2} $
• B. $ \frac{1}{2} $
• C. $ \frac{\sqrt{5}}{2} $
• D. $ \frac{\sqrt{7}}{3} $
• E. $ \sqrt{3} $

Answer 1

Answer: (B)

In $ \Delta CBF,\tan 30{}^\circ =\frac{FC}{b} $
$ FC=\frac{b}{\sqrt{3}} $
$ \Rightarrow $ $ ae=\frac{b}{\sqrt{3}} $
$ \Rightarrow $ $ {{a}^{2}}{{e}^{2}}=\frac{1}{2}[{{a}^{2}}(1-{{e}^{2}})] $
$ \Rightarrow $ $ 4{{e}^{2}}=1 $
$ \Rightarrow $ $ e=\frac{1}{2} $