Problem 50. Two circles with radii $R$ and $r$ touch externally at point $A$. A secant is drawn through point $A$, intersecting the first circle at point $B$ and the second circle at point $C$. A tangent $B D$ is drawn from point $B$ to the second circle. Find the length of $B D$, if $B A=a$.
Solution. From the previous problem, it is known that $A C=\frac{a \cdot r}{R}$, where $R$ and $r$ are the radii of the first and second circles, respectively. Then $B D^{2}=(a+x) a=\left(a+\frac{a \cdot r}{R}\right) a=a^{2} \frac{R+r}{R}$.

Answer. $B D=a \sqrt{\frac{R+r}{R}}$.