Question

If f is the angle between the diameter through any point on a standard ellipse and the normal at the point, then the greatest value of tan f is–

• A. $\frac{2ab}{a^{2} + b^{2}}$
• B. $\frac{a^{2} + b^{2}}{ab}$
• C. $\frac{a^{2} - b^{2}}{2ab}$
• D. $\frac{b^{2}}{a^{2}}$

Answer 1

Answer: (C)

$\frac{a^{2} - b^{2}}{2ab}$

Answer 2

Any point P on ellipse is (a cos q, b sin q)

\ Equation of the diameter CP is y = $\left( \frac{b}{a}\tan\theta \right)$x

The normal to ellipse at P is

ax sec q – by cosec q = a2e2

Slopes of the lines CP and the normal GP are $\frac{b}{a}$tan q and$\frac{a}{b}$tan q

\ tan f = $\frac{\frac{a}{b}\tan\theta - \frac{b}{a}\tan\theta}{1 + \frac{a}{b}\tan\theta.\frac{b}{a}\tan\theta}$=$\frac{\tan\theta}{\sec^{2}\theta}$

=$\frac{a^{2} - b^{2}}{ab}$sin q cos q = $\frac{a^{2} - b^{2}}{2ab}$sin 2q

\ The greatest value of

tan f = $\frac{a^{2} - b^{2}}{2ab}$.1 = $\frac{a^{2} - b^{2}}{2ab}$ .