A line moves so that the sum of the perpendicular drawn to it from

Question

A line moves so that the sum of the perpendicular drawn to it from the points (a,0) and (-a,0) is constant. Show that it always touches a Circle

Answer 1

Let $O$, $A$ and $B$ be the points $(0,0)$, $(-a,0)$ and $(a,0)$ respectively. Let $\ell$ be a line such that the sum of the perpendicular distance from $A$ to $\ell$ and the perpendicular distance from $B$ to $\ell$ is equal to a constant, say $2r$. Here the distances are directed distances. In the figure, $AD$ and $BE$ are in the same direction and hence of have lengths of the same sign. $AF$ and $BG$ are in the opposite directions and have lengths of opposite signs. (i.e., $|AD|+|BE|=2r=|AF|-|BG|$)

The (directed) distance from $O$ to $\ell$ is equal to the mean of the (directed) distance from $A$ to $\ell$ and the (directed) distance from $B$ to $\ell$, which is equal to $r$. So $\ell$ always touches the circle with centre $O$ and radius $r$.