4. If the line $x=\frac{\pi}{4}$ is intercepted by the curve $C:(x-\arcsin a)(x-\arccos a)+(y-\arcsin a)(y+\arccos a)=0$ to form a chord of length $d$, the minimum value of $d$ as $a$ varies is
( A) $\frac{\pi}{4}$
(B) $\frac{\pi}{3}$
(C) $\frac{\pi}{2}$
(D) $\pi$
(C)
4.【Analysis and Solution】From the given conditions, the curve $C$ is a circle with endpoints of its diameter at $P_{1}(\arcsin a, \arcsin a)$ and $P_{2}(\arccos a, -\arccos a)$. The x-coordinate of the center of the circle is $x_{0}=\frac{\arcsin a+\arccos a}{2}=\frac{\pi}{4}$, so the line $x=\frac{\pi}{4}$ passes through the center of the circle, and $d$ is the diameter of the circle. Also, $d^{2}=2\left[(\arcsin a)^{2}+(\arccos a)^{2}\right] \geqslant(\arcsin a+\arccos a)^{2}=\frac{\pi^{2}}{4}$. $\therefore d \geqslant \frac{\pi}{2}$, and when $\arcsin a=\arccos a$, the corresponding $d=\frac{\pi}{2}$. Therefore, the answer is $(\mathrm{C})$.