Chapter 9.2, Problem 60E

To determine

To show: the length of each latus rectum is $\frac{2b^2}{1}$ .

Explanation of Solution

Given information:

An ellipse has two latera recta. Knowing the length of the latera recta is helpful in sketching an ellipse because it yields other points on the curve.

  

Calculation:

Consider the following figure of an ellipse with equation $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ . The eccentricity of the ellipse is $e=\frac{\sqrt{a^2-b^2}}{a}$ .

  

The $y$ coordinate of the point on the ellipse when the $x$ coordinate is $ae$ given by

  $\begin{array}{c}\frac{{\left(ae\right)}^2}{a^2}+\frac{y^2}{b^2}=1\\ \frac{y^2}{b^2}=1-e^2\\ y^2=\left(1-e^2\right)b^2\\ y^2=\left(1-\frac{a^2-b^2}{a^2}\right)b^2\\ y^2=\frac{b^4}{a^2}\\ y=\frac{b^2}{a}\end{array}$

Thus the length of the latus rectum is $\frac{2b^2}{a}$ .