Answer
See explanations.
Work Step by Step
Step 1. Given the ellipse in standard form $\frac{x^2}{a^2} + \frac{y^2}{b^2}=1$, we have $c^2=a^2-b^2$ and the line passing $(c, o)$ perpendicular to the major axis is $x=c$
Step 2. Find the intersection points of the above line with the ellipse: plug-in $x=c$ to the equation, we have
$\frac{c^2}{a^2} + \frac{y^2}{b^2}=1$ and $y^2=b^2(1-\frac{c^2}{a^2})=b^2(\frac{a^2-c^2}{a^2})=\frac{b^4}{a^2}$
Step 3. Find the length of the latus rectum: solve the above equation to get $y=\pm\frac{b^2}{a}$ which are the $y$-values of the intersection points, thus the length of the latus rectum is $y_2-y_1=\frac{2b^2}{a}$
