Question 728·Hard·Math

On the x-axis, $y=0$; on the y-axis, $x=0$. Type both equations. Desmos draws the coordinate axes, which meet at the origin $(0,0)$.

An axis that is tangent touches a circle once; an axis that cuts through it meets it twice. One touch plus two crossings gives three distinct points. If a circle goes through the origin, count that point only once, even though it's on both axes.

Add $(x+4)^2+(y-3)^2=36$. Click its axis intersections: Desmos shows about $(-9.196,0)$, $(1.196,0)$, $(0,-1.472)$, and $(0,7.472)$. That's four different points, not three.

Add $(x-6)^2+(y+2)^2=9$. Desmos shows axis intersections near $(3.764,0)$ and $(8.236,0)$. It misses the $y$-axis, so it meets the axes at only two points.

Add $(x+1)^2+(y-7)^2=25$. Its axis intersections are about $(0,2.101)$ and $(0,11.899)$. It misses the $x$-axis, so this circle also has only two.

Add $(x-3)^2+(y-4)^2=16$. Desmos shows one touch at $(3,0)$ and two more axis intersections near $(0,1.354)$ and $(0,6.646)$. Those are exactly three distinct points, so $(x-3)^2+(y-4)^2=16$ is the requested equation. Choice C.

In a circle's standard form, $(x-h)^2+(y-k)^2=r^2$, the center is $(h,k)$ and the radius is $r$. Its distances to the $x$- and $y$-axes are $|k|$ and $|h|$. If an axis is exactly one radius away, it touches the circle once; if closer, it crosses twice; if farther, it misses.

The circle centered at $(-4,3)$ has radius $6$, so it crosses both axes. The circles centered at $(6,-2)$ with radius $3$ and $(-1,7)$ with radius $5$ each cross one axis and miss the other. The circle centered at $(3,4)$ has radius $4$: the $x$-axis is exactly $4$ units away, while the $y$-axis is only $3$ units away. It touches one axis and crosses the other, so $(x-3)^2+(y-4)^2=16$ gives three distinct points. Choice C.