Question

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

• A. Depends upon foci
• B. Is constant
• C. Depends upon the tangent
• D. None of the above

Answer 1

The correct option is B Is constant

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 equation (ii) be $SM$ and $S^{{}^{\prime }}N$.
Then, $SM\cdot S^{{}^{\prime }}N=\frac{-1}{\frac{{\cos }^2\theta }{a^2}+\frac{{\sin }^2\theta }{b^2}}(e^2{\cos }^2\theta -1)$
$\Rightarrow SM\cdot S^{{}^{\prime }}N=b^2=$ Constant