Question 672·Hard·Math

Type the circle equation into Desmos exactly as given, without adding $y=$. Click the circle. Desmos marks its highest point, $(3,0)$, and lowest point, $(3,-8)$. These mark a vertical diameter; they aren’t the endpoints $P$ and $Q$ on the slanted line.

The top and bottom of a circle sit directly above and below its center. Both marked points have $x=3$, and $-4$ is halfway between their $y$-coordinates, $0$ and $-8$. So the center is $(3,-4)$.

Add $y=-x-1$ on a second Desmos line. At the center’s $x$-coordinate, $x=3$, the line gives $y=-3-1=-4$. That matches the center’s $y$-coordinate, so the line passes through $(3,-4)$.

A chord is a segment joining two points on a circle. Because this chord passes through the center, it’s a diameter. Every diameter of the same circle has the same length. The vertical one runs from $y=0$ to $y=-8$, so its length is $0-(-8)=8$. The length of $\overline{PQ}$ is $8$. Choice C.