Question

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

• A. $ {{a}^{2}}+{{b}^{2}} $
• B. $ \sqrt{{{a}^{2}}+{{b}^{2}}} $
• C. $ ({{a}^{2}}+{{b}^{2}}) $
• D. None of these

Answer 1

Answer: (D)

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 $
$\cos ^{2} \theta+\sin ^{2} \theta=\frac{a^{2}}{l^{2}}+\frac{b^{2}}{l^{2}}=1 \Rightarrow l^{2}=a^{2}+b^{2} \therefore l=\sqrt{a^{2}+b^{2}}$