How to get the radius of an ellipse at a specific angle by knowing its semi-major and semi-minor axes?

Question

How to get the radius of an ellipse at a specific angle by knowing its semi-major and semi-minor axes?

Please take a look at this picture :

Answer 1

The polar form of the equation for an ellipse with "horizontal" semi-axis $a$ and "vertical" semi-axis $b$ is

$$r = \frac{ab}{\sqrt{a^2\sin^2\theta+b^2\cos^2\theta}}$$

Here, $\theta$ represents the angle measured from the horizontal axis ($30.5^\circ$ in your case), and $r$ is the distance from the center to the point in question (the radius you seek).

Answer 2

So, assuming the major axis to be the $Y$ axis and centre $(0,0)$

the equation of the ellipse is $$\frac{x^2}{(2.23)^2}+\frac{y^2}{(3.05)^2}=1$$

and the equation of the line of $r$ is $\frac yx=\tan(30.5-90)^\circ$

Now, if $(h,k)$ is one the points of intersection $r=\sqrt{h^2+k^2}$