Conic Sections Question 303
Question: If the angle between the straight lines joining the foci to an extremity of minor axis in an ellipse be $ 90{}^\circ $ ; then the eccentricity of the ellipse is
Options:
A) $ \frac{1}{2} $
B) $ \frac{1}{\sqrt{3}} $
C) $ \frac{1}{\sqrt{2}} $
D) $ \frac{1}{3} $
Show Answer
Answer:
Correct Answer: C
Solution:
[c] From standard equation of ellipse $ \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 $ , the Co-ordinates of foci are $ S(ae,0) $ And $ S’(-ae,0). $ Co-ordinate of an extremity of the minor axis is B (0, b), Now slope of straight line $ BS=\frac{b-a}{0-ae}=\frac{b}{-ae}=m_1 $ And slope of straight line $ BS’=\frac{b-a}{0-(-ae)}=\frac{b}{ae}=m_2 $ $ \because SB\bot BS’, $ so $ m_1.m_2=-1 $ or $ \frac{b}{-ae}\times \frac{b}{ae}=-1; $ or $ \frac{b^{2}}{a^{2}}=e^{2} $ But $ b^{2}=a^{2}(1-e^{2}); $ or $ \frac{b^{2}}{a^{2}}=1-e^{2}; $ or $ e^{2}=1-e^{2}. $ Or $ 2e^{2}=1; $ or $ e=\frac{1}{\sqrt{2}}. $
BETA
