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Q.
If any tangent to the ellipse $\frac{x^{2}}{16}+\frac{y^{2}}{9}=1$ intercepts equal lengths $l$ on the axes, then $l=$
Conic Sections
Solution:
The equation of any tangent to the given ellipse is
$\frac{x}{4} \cos \theta+\frac{y}{3} \sin \theta=1$
The tangent meets $x$-axis at $A\left(\frac{4}{\cos \theta}, 0\right)$ and $y$-axis at $\left(0, \frac{3}{\sin \theta}\right)$
Given : $ \frac{4}{\cos \theta}=l=\frac{3}{\sin \theta} $
$\Rightarrow \cos \theta=\frac{4}{l}$ and $\sin \theta=\frac{3}{l}$
$\Rightarrow \cos ^{2} \theta+\sin ^{2} \theta=\frac{16}{l^{2}}+\frac{9}{l^{2}} $
$\Rightarrow l^{2}=25$.
$ \therefore l=5$