Length of chord on ellipse

Question

Suppose I have an ellipse centered at the origin, preferably expressed in its matrix form, and I want to know the chord length of a segment that passes through the origin with the endpoints at the ellipse, of which I know the angle theta. How can I compute it? Thank you!

Answer 1

Assume the parametric equation of the ellipse to be $(a\cos\theta, b\sin\theta)$ where a is the semi-major axis and b is the semi-minor axis. Since an ellipse is a central curve, the origin bisects the chord and therefore the length of the chord is twice its distance from the origin, i.e., $$2\sqrt{(a\cos \theta)^{2}+(b\sin \theta)^{2}}$$

Answer 2

If your chord makes an angle $\phi$ with the $x$-axis, then its length is $$ \frac{2ab}{\sqrt{ {b^2}\cos^2{\phi} + {a^2}\sin^2{\phi} }} $$ See this answer for a bit more detail.

The angle $\theta$ used in apurv's answer is not the angle between the chord and the $x$-axis. That's probably the reason for his question, where he asked you what you mean by "theta".