A) \[\frac{1}{2}\]
B) \[\frac{1}{\sqrt{3}}\]
C) \[\frac{1}{\sqrt{2}}\]
D) \[\frac{1}{3}\]
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 \[2{{e}^{2}}=1;\] or \[e=\frac{1}{\sqrt{2}}.\] |
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