Question

The product of the perpendiculars drawn from the foci upon any tangent to an ellipse

• A. depends upon foci
• B. is onstant
• C. depends upon the tangen
• D. None of the above

Answer 1

Answer: (B)

Let equation of an ellipse is
$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$... (i)

Its foci are $S(ae,0)$ and $S^{\prime }(-ae,0)$. The equation of tangent at any point $(a\cos \theta ,b\sin \theta )$ to ellipse is
$\frac{x}{a}\cos \theta +\frac{y}{b}\sin \theta =1$ ...(ii)
Let the perpendicular from $S$ and $S^{\prime }$ upon Eq. (ii) be $SM$ and $S^{\prime }N$.
Then, $SM\cdot S^{\prime }N=\frac{-1}{\frac{{\cos }^2\theta }{a^2}+\frac{{\sin }^2\theta }{b^2}}\left(e^2{\cos }^2\theta -1\right)$
$\Rightarrow SM\cdot S^{\prime }N=b^2=$ Constant