14.3. (Curie, USA, 77). Through a point located at a distance $k<1$ from the center of a circle of radius 1, a pair of perpendicular chords is drawn. Find the greatest and the least value of the sum of their lengths.
14.3. Let through point $A$, located at a distance $k$ from the center $O$ of the circle, perpendicular chords $K L$ and $M N$ be drawn. Drop perpendiculars $O B$ and $O C$ to the chords $K L$ and $M N$ respectively and denote $\angle A O B=\alpha$ (Fig. 93). Then

$$
\begin{aligned}
& K L=2 B L=2 \sqrt{1-k^{2} \cos ^{2} \alpha} \\
& M N=2 M C=2 \sqrt{1-k^{2} \sin ^{2} \alpha}
\end{aligned}
$$

and

$$
\begin{aligned}
(K L+M N)^{2} & =8-4 k^{2}\left(\cos ^{2} \alpha \dot{+} \sin ^{2} \alpha\right)+ \\
& +8 \sqrt{1-k^{2}\left(\cos ^{2} \alpha+\sin ^{2} \alpha\right)+k^{4} \cos ^{2} \alpha \sin ^{2} \alpha}= \\
& =8-4 k^{2}+4 \sqrt{4-4 k^{2}+k^{4} \sin ^{2} 2 \alpha}
\end{aligned}
$$

The last expression is maximal when $\sin 2 \alpha=1$, i.e., when $\alpha=45^{\circ}$, and minimal when $\sin 2 \alpha=0$, i.e., when $\alpha=0^{\circ}$ or $\alpha=90^{\circ}$. Therefore, the maximum value of $K L+M N$ is

$$
\sqrt{8-4 k^{2}+4\left(2-k^{2}\right)}=2 \sqrt{4-2 k^{2}},
$$

and the minimum value is

$$
\sqrt{8-4 k^{2}+8 \sqrt{1-k^{2}}}=2\left(1+\sqrt{1-k^{2}}\right)
$$