Mongolian Mathematical Olympiad

The problem is clear if the two planes are parallel, thus we assume that they intersect on line $l$. Let $r$ denote the radius of the sphere and let $O$ denote the center of the sphere. Let $a>r$ denote the distance from $O$ to $l$.

A sphere with center $P$ and radius $s$ satisfies the condition of the problem iff $$r+s=OP\text{and}\frac{r}{a}=\frac{s}{b}.$$ We may assume that $r=1$. Consider a coordinate system where $l$ is the $x$-axis and $O(0,a,0)$. Then $P(x,y,z)$ satisfies $y=b>0$ and $z=0$. Since $a>1$, the equation $$x^2+(y-a{)}^2={\left(1+\frac{y}{a}\right)}^2$$ gives an ellipse.