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 $OP_i = d_i$ for $i = 1, 2, 3$. As the tangent line is perpendicular to the radius drawn to the tangent point, $OP_1P_3$ and $OP_2P_3$ are right triangles with hypothenuses $OP_1$ and $OP_2$, respectively. Hence
$$P_1P_2 = P_1P_3 + P_2P_3 = \sqrt{d_1^2 - d_3^2} + \sqrt{d_2^2 - d_3^2}.$$
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, $2\sqrt{(2r+h+l)(h-l)}$.