2. (mathematics) Segments $A B$ and $C D$, with lengths 3 and 4 respectively, are chords of a circle with radius 5. Points $M$ and $N$ are located on them such that $A M: M B=1: 2, \quad C N: N D=1: 3$. Find the smallest and largest possible lengths of segment $M N$.
Answer: $d_{\min }=\sqrt{23}-\sqrt{22} \quad d_{\max }=\sqrt{23}+\sqrt{22}$

Solution. The lengths of segments $A B=a, C D=b, \quad A M: M B=p: q$ $C N: N D=m: n$. Points $P, Q$ are the midpoints of segments $A B$ and $D C$. We will prove that points $M$ and $N$ lie on two concentric circles for any position of segments $A B$ and $C D$ and find their

![](https://cdn.mathpix.com/cropped/2024_05_06_7d3a82aa7ed091c0d22dg-2.jpg?height=568&width=579&top_left_y=2035&top_left_x=1361)
radii.

$$
\begin{gathered}
C N=\frac{b m}{m+n} \cdot N Q=\frac{b}{2}-\frac{b m}{m+n}=\frac{b(n-m)}{2(m+n)} \\
O N^{2}=N Q^{2}+O Q^{2}=N Q^{2}+R^{2}-\frac{b^{2}}{4}=R^{2}-\frac{m n}{(m+n)^{2}} b^{2}
\end{gathered}
$$

Similarly,

$$
O M^{2}=R^{2}-\frac{p q}{(p+q)^{2}} a^{2}
$$

Thus, the distances from points $M$ and $N$ to the center of the circle are constant and do not depend on the position of segments $A B$ and $C D$ on this circle, i.e., points $M$ and $N$ lie on circles with center at point $O$ with radii

$$
r_{1}=\sqrt{R^{2}-\frac{p q}{(p+q)^{2}} a^{2}} \text { and } r_{2}=\sqrt{R^{2}-\frac{m n}{(m+n)^{2}} b^{2}}
$$

![](https://cdn.mathpix.com/cropped/2024_05_06_7d3a82aa7ed091c0d22dg-3.jpg?height=579&width=651&top_left_y=567&top_left_x=1271)

respectively. Then the minimum distance between points $M$ and $N$ is $\left|r_{1}-r_{2}\right|$, and the maximum distance is $r_{1}+r_{2}$.

In the case $p=1, q=2, m=1, n=3, a=3, b=4, R=5$