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. $\pi$
Solve:
$(x-\arcsin a)(x-\arccos a)+(y-\arcsin a)(y+\arccos a)=0$
$\Rightarrow x^{2}+y^{2}-\frac{\pi}{2} x+\left(\frac{\pi}{2}-2 \arcsin a\right) y=0$, so the line $x=\frac{\pi}{4}$ passes through the center of the circle,
thus the length of the chord intercepted by the line $x=\frac{\pi}{4}$ on the curve $C$ is $d=2 r$.
Therefore, $d^{2}=4 r^{2}=\frac{\pi^{2}}{4}+\left(\frac{\pi}{2}-2 \arcsin a\right)^{2} \geqslant \frac{\pi^{2}}{4} \Rightarrow d \geqslant \frac{\pi}{2}$. So the answer is $C$.