11.14. On the hypotenuse $AB$ of a right triangle $ABC$, a point $X$ is taken; $M$ and $N$ are its projections on the legs $AC$ and $BC$.

a) For which position of point $X$ will the length of segment $MN$ be the smallest?

b) For which position of point $X$ will the area of quadrilateral $CMXN$ be the largest?
11.14. a) Since $C M X N$ is a rectangle, then $M N = C X$. Therefore, the length of the segment $M N$ will be the smallest if $C X$ is the height.

b) Let $S_{A B C} = S$. Then $S_{A M X} = A X^{2} \cdot S / A B^{2}$ and $S_{B N X} = B X^{2} \cdot S / A B^{2}$. Since $A X^{2} + B X^{2} \geqslant A B^{2} / 2$ (with equality only if $X$ is the midpoint of segment $A B$), then $S_{C M X N} = S - S_{A M X} - S_{B N X} \leqslant S / 2$. The area of quadrilateral $C M X N$ will be the largest if $X$ is the midpoint of side $A B$.