Estonian Mathematical Olympiad

Fig. 9

If the distance is maximal, the line connecting these points must be a tangent of the planet, otherwise one could increase the distance by pushing the points along the surface of the planet farther away. Let the centre be $O$; let the two points under consideration be $P_1$ and $P_2$ with the tangent point $P_3$ between them (Fig. 9). Denote $O{P}_i=d_i$ for $i=1,2,3$. As the tangent line is perpendicular to the radius drawn to the tangent point, $O{P}_1{P}_3$ and $O{P}_2{P}_3$ are right triangles with hypothenuses $O{P}_1$ and $O{P}_2$, respectively. Hence $$P_1{P}_2=P_1{P}_3+P_2{P}_3=\sqrt{d_1^2-d_3^2}+\sqrt{d_2^2-d_3^2}.$$ The value of this expression is maximal if $d_1$ and $d_2$ are as large as possible and $d_3$ is as small as possible, i.e., $d_1=d_2=r+h$ and $d_3=r+l$. Substituting these values gives the desired distance $2\sqrt{(r+h{)}^2-(r+l{)}^2}$, or equivalently, $2\sqrt{(2r+h+l)(h-l)}$.