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  • If any tangent to the ellipse <img alt="\mathrm{\frac{x^2}{a^2}+\frac{y^2}{b^2}=1}" src="http

If any tangent to the ellipse \mathrm{\frac{x^2}{a^2}+\frac{y^2}{b^2}=1} intercepts equal lengths l on the axes, then l= 

Option: 1

\mathrm{a^2+b^2}


Option: 2

\mathrm{\sqrt{a^2+b^2}}


Option: 3

\mathrm{\left(a^2+b^2\right)^2}


Option: 4

None of these


Answers (1)

best_answer

The equation of any tangent to the given ellipse is \mathrm{\frac{x}{a} \cos \theta+\frac{y}{b} \sin \theta=1}
This line meets the coordinate axes at\mathrm{P\left(\frac{a}{\cos \theta}, 0\right) \quad Q\left(0, \frac{b}{\sin \theta}\right)}
\mathrm{\begin{aligned} & \therefore \quad \frac{a}{\cos \theta}=I=\frac{b}{\sin \theta} \Rightarrow \cos \theta=\frac{a}{l} \text { and } \sin \theta=\frac{b}{l} \Rightarrow \cos ^2 \theta+\sin ^2 \theta=\frac{a^2}{l^2}+\frac{b^2}{l^2} \Rightarrow I^2=a^2+b^2 \Rightarrow \\ & I=\sqrt{a^2+b^2} \end{aligned}}

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Rishabh

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