Q. 52 - Calculus

Question

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

Step-by-Step Solution

Answer

Hence proved.

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.