Solve the following equation:

$$
\frac{\sqrt{a+x}+\sqrt{a-x}}{\sqrt{a+x}-\sqrt{a-x}}=\sqrt{b} .
$$
Our equation can also be written as:

$$
\sqrt{a+x}(\sqrt{b}-1)=\sqrt{a-x}(\sqrt{b}+1) .
$$

Squaring both sides:

$$
(a+x)(b+1-2 \sqrt{b})=(a-x)(b+1+2 \sqrt{b})
$$

from which

$$
x=\frac{2 a \sqrt{b}}{b+1}
$$

(Fried Ernö, Budapest).)