ellipse chord length along its axis.

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how to determine the position in an ellipse, where the chord length is equal to its minor axis and perpendicular to the major axis? Is there any equation for it?

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You want to find $x_0$ such that the $y_0$ satisfying $$\frac{x_0^2}{a^2} + \frac{y_0^2}{b^2} = 1$$ is equal to $a.$ You should be able to solve this.