10.5. A circle of unit radius with center 0 is given. Point $A$ is at a distance $a$ from the center ( $0<\mathrm{a}<1$ ). Through point $A$, all possible chords $M N$ are drawn. a) Find the length of the shortest chord $M$. b) Find the maximum area of triangle $OMN$.
Answer: a) $2 \sqrt{1-a^{2}}$; b) when $a \geq \frac{\sqrt{2}}{2}$ the maximum area is $\frac{1}{2}$; when $a < \frac{\sqrt{2}}{2}$, the maximum area is $2 \sqrt{1-a^{2}} \cdot a$. The maximum value of $\sin \alpha$ is reached when $\alpha = \alpha_{0}$, because $\alpha_{0} > \frac{\pi}{2}$ and due to the monotonic decrease of the function $\sin x$ in the second quadrant, the maximum value of $\sin \alpha$ will be at $\alpha=\alpha_{0}$.