What is the maximum number of common tangents that two ellipses can have if they share one focus?
It is known that if we connect the point of tangency $E$ of a tangent $e$ of an ellipse with the foci $F_{1}$ and $F_{2}$ of the ellipse, then the external angle bisector of the thus obtained focal radii is the tangent $e$ (Figure 1). This implies that if we reflect the focus $F_{2}$ over all the tangents of the ellipse, the set of reflections will be the circle circumscribed around $F_{1}$ with radius $2a$ - where $2a$ is the length of the major axis of the ellipse - because if $F_{2}^{\prime}$ is the reflection of $F_{2}$ over $e$, then $F_{2}^{\prime} F_{1} = F_{2}^{\prime} E + E F_{1} = F_{2} E + F_{1} E = 2a$, since the angles are equal, the points $F_{2}^{\prime}, E$, and $F_{1}$ lie on a straight line, and by reflection, $F_{2}^{\prime} E = F_{2} E$. (The detailed proof of these statements can be found, for example, in the theorems 42.1 and 42.2 of György Hajós's book "Introduction to Geometry".)

Let the common focus of the two ellipses in the problem be denoted by $F_{2}$, the other foci by $F_{1}$ and $F_{3}$, and the lengths of the major axes by $2a_{1}$ and $2a_{3}$, respectively. According to the previous, if we reflect $F_{2}$ over a common tangent of the two ellipses, the resulting point $F_{2}^{\prime}$ will lie on both the circle with center $F_{1}$ and radius $2a_{1}$, and the circle with center $F_{3}$ and radius $2a_{3}$. If the two circles do not coincide (i.e., if the two ellipses are different), then they have at most two intersection points, so the two ellipses can have at most two common tangents. If $P$ is a common point of the two circles, then the perpendicular bisector of the segment $P F_{2}$ is a tangent to both ellipses (Figure 2). Since the number of common points of two different circles is 0, 1, or 2, if two ellipses have a common focus but are different, the number of their common tangents can be 0, 1, or 2, and all three cases can occur (Figure 3).

Based on the work of Róbert Belső (Bonyhád, Petőfi S. Gymnasium, 9th grade)
![](https://cdn.mathpix.com/cropped/2024_05_02_7a54d0c3f8a77b7dd6c6g-1.jpg?height=1590&width=596&top_left_y=832&top_left_x=754)

![](https://cdn.mathpix.com/cropped/2024_05_02_7a54d0c3f8a77b7dd6c6g-2.jpg?height=183&width=348&top_left_y=188&top_left_x=877)