Question

Show that the eccentricity satisfies the equation$B^2=$$A^2\left(1-e^2\right)$.

Short answer

Hence proved.

Step-by-step solution

Step 1. Given information.

It is given$B^2=A^2\left(1-e^2\right)$.

Step 2. Conclusion.

Now,

$$\begin{gathered} B^2=A^2\left(1-{\left(\frac{\sqrt{A^2-B^2}}{A}\right)}^2\right)\left[\text{since by substituting}e=\frac{\sqrt{A^2-B^2}}{A}\right] \\ =A^2\left(1-\frac{{\left(\sqrt{A^2-B^2}\right)}^2}{A^2}\right) \\ =A^2\left(\frac{A^2-\left(A^2-B^2\right)}{A^2}\right) \\ =A^2\left(\frac{A^2-A^2+B^2}{A^2}\right) \\ =A^2\left(\frac{B^2}{A^2}\right) \\ {B}^2=B^2 \end{gathered}$$

Hence proved.