Where does a chord of an Ellipse equal to the length of the minor axis but running parallel to the major axis cross the minor axis.

Question

Given an Ellipse, I need to know where a chord equal to the length of the minor axis but running parallel to the major axis cross the minor axis.

Answer 1

$\frac {x^2}{a^2} + \frac {y^2}{b^2} = 1$

Let $a>b$

the minor axis has length $2b$

We have a chord with $x$ coordinates $-b,b$ and we need to find the y coordinates.

$\frac {b^2}{a^2} + \frac {y^2}{b^2} = 1\\ \frac {y^2}{b^2} = 1 - \frac {b^2}{a^2}\\ y^2 = b^2(\frac {a^2 - b^2}{a^2})\\ y = \pm\frac {b}{a}\sqrt {a^2 - b^2}$