A) \[{{a}^{2}}+{{b}^{2}}\]
B) \[\sqrt{{{a}^{2}}+{{b}^{2}}}\]
C) \[({{a}^{2}}+{{b}^{2}})\]
D) None of these
Correct Answer: D
Solution :
The equation of tangent to the given ellipse at point \[P(a\cos \theta ,\,\,b\sin \theta )\]is\[\frac{x}{a}\cos \theta +\frac{y}{b}\sin \theta =1\] Intercept of line on the axes are\[\frac{a}{\cos \theta }\]and\[\frac{b}{\sin \theta }\]. Given that,\[\frac{a}{\cos \theta }=\frac{b}{\sin \theta }=l\] \[\Rightarrow \] \[\cos \theta =\frac{a}{l}\] and \[\sin \theta =\frac{b}{l}\] \[\Rightarrow \] \[{{\cos }^{2}}\theta +{{\sin }^{2}}\theta =\frac{{{a}^{2}}}{{{l}^{2}}}+\frac{{{b}^{2}}}{{{l}^{2}}}=1\] \[\Rightarrow \] \[{{l}^{2}}={{a}^{2}}+{{b}^{2}}\] \[\therefore \] \[l=\sqrt{{{a}^{2}}+{{b}^{2}}}\]
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