I've been beating my head against a wall for several days trying to figure this out. It doesn't seem to be a documented problem on the internet anywhere.
I have a horizontal line segment of known length (R). This line is tangent to an ellipse and one endpoint contacts it at the topmost point of the ellipse. At the end of line segment R is another line segment of unknown length. This line segment is at at a known angle off vertical (Θ). This line segment is tangent to the ellipse as well. I also know the "aspect ratio" of the ellipse (B / A).
A is the major dimension and B is the minor dimension. A is parallel to R (i.e. the ellipse is wider than it is tall). A and B are half the overall width and height of the ellipse, respectively.
What I need to calculate are the dimensions of the ellipse (lengths A and B) as well as the location of the contact point on the ellipse of the angled line segment (lengths H and S). Is there a formula for this?
The blue lines in the diagram represent the connection lines from the tangent point to the focii.