Question

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

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

Answer 1

Answer: (B)

$5$

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 $A(\frac{4}{\cos \theta },0)$

and y-axis at $(0,\frac{3}{\sin \theta })$.

Given: $\frac{4}{\cos \theta }=l=\frac{3}{\sin \theta }\Rightarrow \cos \theta =\frac{4}{l}$ and $\sin \theta =\frac{3}{l}$

$\Rightarrow {\cos }^2\theta +{\sin }^2\theta =\frac{16}{l^2}+\frac{9}{l^2}$

$\Rightarrow l^2=25\Rightarrow l=5$