Question

If any tangent to the ellipse $\frac{x^2}{16}+\frac{y^2}{9}=1$ intercepts equal lengths $l$ on the axes, then $l=$

• A. 3
• B. 5
• C. $\sqrt{5}$
• D. none of these

Answer 1

Answer: (B)

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 $\sin \theta =\frac{3}{l}$
$\Rightarrow {\cos }^2\theta +{\sin }^2\theta =\frac{16}{l^2}+\frac{9}{l^2}$
$\Rightarrow l^2=25$.
$\therefore l=5$