IMC 2008 · Day 2 · P8

It is well known that an ellipse might be defined by a focus (a point) and a directrix (a straight line), as a locus of points such that the distance to the focus divided by the distance to the directrix is equal to a given number $e < 1$. So, if a point $X$ belongs to both ellipses with the same focus $F$ and directrices $l_1$, $l_2$, then $e_1 \cdot l_1 X = FX = e_2 \cdot l_2 X$ (here we denote by $l_1 X$, $l_2 X$ distances between the corresponding line and the point $X$). The equation $e_1 \cdot l_1 X = e_2 \cdot l_2 X$ defines two lines, whose equations are linear combinations with coefficients $e_1, \pm e_2$ of the normalized equations of lines $l_1, l_2$ but of those two only one is relevant, since $X$ and $F$ should lie on the same side of each directrix. So, we have that all possible points lie on one line. The intersection of a line and an ellipse consists of at most two points.