回復 6# 阿光 的帖子
第10題
作\(\overline{CD}\)垂直x軸於D
令∠\(CAD=\theta \quad \left( 0\le \theta \le \frac{\pi }{2} \right)\),則∠\(ABO=\theta \)
\(\begin{align}
& \overline{OA}=\overline{CD}=a\sin \theta ,\overline{AD}=a\cos \theta \\
& C\left( a\left( \sin \theta +\cos \theta \right),a\sin \theta \right) \\
& \overline{OC}=a\sqrt{{{\left( \sin \theta +\cos \theta \right)}^{2}}+{{\sin }^{2}}\theta } \\
& =a\sqrt{1+\sin 2\theta +{{\sin }^{2}}\theta } \\
& =a\sqrt{\sin 2\theta -\frac{1}{2}\cos 2\theta +\frac{3}{2}} \\
\end{align}\)
故\(\overline{OC}\)最大值為\(\sqrt{\frac{\sqrt{5}}{2}+\frac{3}{2}}\ a\),最小值為\(a\)