Calculating a Point that lies on an Ellipse given an Angle

I need to find a point (A on this diagram) given the center point of the ellipse as well as an angle. I've been melting my brain all day (as well as searching through questions here) testing out different equations. What's the best way to do this?

I intend to grab point A at $225^o$ as well as another point at approximately $250^o$ using the same math. These need to be fetched regardless of elliptic width and height.

Solution 1:

If the ellipse is centered at the origin, the equation of the ellipse is $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$. The equation of the line is $y=x\tan \theta $ So you have $\frac{x^2}{a^2}+\frac{(x\tan \theta )^2}{b^2}=1$ or $x=\pm \frac{ab}{\sqrt{b^2+a^2(\tan \theta)^2}}$ where the sign is + if $ -\pi/2 \lt \theta \lt \pi/2$

Solution 2:

You can also use parametric equations:

$$x=a\cos(\theta)$$ $$y=b\sin(\theta)$$

Where $a$ is the radius on the horizontal axis, and $b$ is the radius on the vertical axis.