A) \[{{\tan }^{-1}}\left( \pm \frac{ae}{b} \right)\]
B) \[{{\tan }^{-1}}\left( \pm \frac{be}{a} \right)\]
C) \[{{\tan }^{-1}}\left( \pm \frac{b}{ae} \right)\]
D) \[{{\tan }^{-1}}\left( \pm \frac{a}{be} \right)\]
Correct Answer: C
Solution :
Coordinates of any point on the ellipse \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\] whose eccentric angle is \[\theta \]are \[(a\cos \theta ,\,\,b\sin \theta ).\] The coordinates of the end points of latus recta are \[\left( ae,\,\pm \frac{{{b}^{2}}}{a} \right).\] \[\therefore a\cos \theta =ae\] and \[b\sin \theta =\pm \frac{{{b}^{2}}}{a}\] Þ \[\tan \theta =\pm \frac{b}{ae}\Rightarrow \theta ={{\tan }^{-1}}\left( \pm \frac{b}{ae} \right)\].
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