Semi-minor Axis of Circumscribed Ellipse

Question

I met a problem in calculation. When the ellipse is tangential to the circle (there is only one point touched, see what shown in the graph) and semi-major axis and radius is given, if it is possible to calculate the semi-minor axis of the ellipse. I have no clue how to solve that. Could you plz give some suggestions

Answer 1

Given the length of the semi-major axis $a$ and the radius $r$ of the circle inscribed at the vertex $(a ,0)$ of the ellipse, the semi-major axis $b$ might be anything between $\sqrt{a\cdot r}$ and $a$.

The minimal value $b_{\min}=\sqrt{a\cdot r}$ corresponds to the curvature of the ellipse at the point of tangency.

Answer 2

The radius of curvature of the ellipse with eqution

$$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\tag{1}$$

is given by (https://math.stackexchange.com/q/1452057)

$$R_{(x,y)}=\frac{1}{ab} \left (\tfrac{b}{a} x)^2 + (\tfrac{a}{b} y)^2\right)^{3/2}\tag{2}$$

The radius of curvature at the right end of the ellipse $(x,y)=(a,0)$.

$$R_{(a,0)}=\frac{1}{ab} \left (\tfrac{b}{a} a)^2\right)^{3/2}=\frac{b^2}{a}\tag{3}$$

Therefore, if you know $a$ and $R_{(a,0)}$, relationship (3) will give you $b$.