Find the length of the chord that the line $y=3 x$ cuts off from the circle $(x+1)^{2}+(y-2)^{2}=25$.

#
Let's find the coordinates of the points of intersection \(A\left(x_{1} ; y_{1}\right)\) and \(B\left(x_{2} ; y_{2}\right)\) of the given line and circle. For this, we will solve the system of equations

\[
\left\{\begin{array}{l}
y=3 x \\
(x+1)^{2}+(y-2)^{2}=25
\end{array}\right.
\]

We obtain: \(x_{1}=-1, y_{1}=-3, x_{2}=2, y_{2}=6\).

Using the formula for the distance between two points

\[
A B=\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}}=\sqrt{(2-(-1))^{2}+(6-(-3))^{2}}=\sqrt{9+81}=3 \sqrt{10}
\]

## Answer

\(3 \sqrt{10}\).