## Task $5 / 85$

Given a circle with center $M$, radius $r$, and a chord $A B=s$. By how much must $A B$ be extended beyond $B$ so that the tangent from the endpoint $E$ of the extension to the circle has length $t$?
Let $B E=x$. According to the tangent-secant theorem, we have

$$
t^{2}=(x+s) x \quad \text { so } \quad x^{3}+x s-t^{2}=0 \rightarrow x=0.5\left(\sqrt{s^{2}+4 t^{2}}-s\right)
$$

(due to $x>0$, the second solution is not feasible).