4.3. Two balls of one radius and two of another are arranged so that each ball touches three others and a given plane. Find the ratio of the radii of the balls.
4.3. Let $A$ and $C$ be the points of tangency of spheres of radius $R$ with the plane; $B$ and $D$ be the points of tangency of spheres of radius $r$ with the plane. According to problem 4.1, $A B = B C = C D = A D = 2 \sqrt{R r}$, so $A B C D$ is a rhombus; its diagonals are equal to $2 R$ and $2 r$. Therefore, $R^{2} + r^{2} = 4 R r$, i.e., $R = (2 \pm \sqrt{3}) r$. Thus, the ratio of the larger radius to the smaller one is $2 + \sqrt{3}$.