finding eccentricity of ellipse??

Question

If the tangent at any point of the ellipse make an angle α with major axis and an angle β with focal radius of the point of contact then show that the eccentricity of the ellipse is given by e=cosβ/cosα..

Answer 1

Let the ellipse be $$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1.$$ where $a\gt b.$

Also, suppose that the point of contact $P(a\cos\theta, b\sin\theta)$ is in the first quadrant. And let $Q$ be the intersection point of the tangent and $x$-axis.

Since the tangent is represented as $$\frac{x\cos\theta}{a}+\frac{y\sin\theta}{b}=1,$$ we can find the coordinate of the point $H$ which is an intersection point of the tangent and its perpendicular line from $F(\sqrt{a^2-b^2},0)$.

Hence, we can get $$\cos\alpha=\frac{GH}{FQ}.$$

On the other hand, we get $$\cos\beta=\frac{{BF}^2+{BF^\prime}^2-{FF^\prime}^2}{2\cdot BF\times BF^\prime}$$ where $F^\prime (-\sqrt{a^2-b^2},0).$

So, all we need is to prove the following equation independently to $\theta.$ $$e=\frac{\sqrt{a^2-b^2}}{a}=\frac{\cos\beta}{\cos\alpha}.$$