The distance from the center of a circle, which is tangent to a ellipse $x^2/a^2+y^2/b^2=1$ and two parallel tangent lines of the ellipse is $a+b$

Question

Consider the following problem:

Let $E$ be the ellipse $x^2/a^2+y^2/b^2=1$ with $a>b$. Consider two tangent lines on $E$ which are parallel, say, $r$ and $s$. Let $C$ be a circle, which is tangent to $r$, $s$, and the ellipse. Show that the distance from the center of the circle $C$ to the center of the $E$ is $a+b$.

My question is: does anyone know the origin of this problem? Does it have a name?

Answer 1

According to problems.ru, this problem appears in Prasolov's "Problems in plane geometry", 2001 edition.

A 2006 edition (in Russian) is available at the publisher web site.

This problem appears on page 586 as problem #31.25:

Translation: Two parallel lines $l_1$ and $l_2$ are drawn tangent to an ellipse with the center at the point $O$. A circle with the center at the point $O_1$ touches the ellipse (on the outside) and the lines $l_1$ and $l_2$. Prove that the length of the segment $OO_1$ equals the sum of the half-axes of the ellipse.

According to the introduction to the fifth edition, section 31 ("Ellipse, parabola, hyperbola") appeared in the first edition (1986) but was omitted in editions 2 through 4.