A) Focus
B) Centre
C) End of the major axes
D) End of minor axes
Correct Answer: B
Solution :
Let \[P(a\,\,\cos {{\theta }_{1}},\,b\sin {{\theta }_{1}})\] and \[Q\,(a\,\cos {{\theta }_{2}},\,b\sin {{\theta }_{2}})\]be two points on the ellipse. Then |
\[{{m}_{1}}=\] Slope of OP (O is centre) \[=\frac{b}{a}\,\tan {{\theta }_{1}};\] |
\[{{m}_{2}}=\] Slope of \[OQ=\frac{b}{a}\tan {{\theta }_{2}}\] |
Since, \[{{m}_{1}}{{m}_{2}}=\frac{{{b}^{2}}}{{{a}^{2}}}\,\tan {{\theta }_{1}}\,\tan {{\theta }_{2}}\] |
\[=\frac{{{b}^{2}}}{{{a}^{2}}}\left( -\frac{{{a}^{2}}}{{{b}^{2}}} \right)=-1\,\,(\because \,\,\tan {{\theta }_{1}}\,\tan {{\theta }_{2}}=\frac{-{{a}^{2}}}{{{b}^{2}}})\] |
\[\therefore \,\,\angle POQ=90{}^\circ .\] Hence PQ makes a right angle at the centre of the ellipse. |
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