Question 128·Hard·Geometry and Trigonometry

For a circle with center $(h,k)$, how far is that center from the x-axis (line $y=0$) and from the y-axis (line $x=0$)? How can you express those distances using $h$ and $k$?

If the circle’s radius is larger than the distance from its center to an axis, how many times will it cross that axis? What if the radius equals that distance? What if the radius is smaller?

You need a total of 3 points where the circle lies on the axes. That means one axis must contribute 1 point and the other must contribute 2 points. Which equation produces that combination when you compare its radius to its distances from the axes?

In Desmos, type each equation from the answer choices on its own line (for example, (x + 4)^2 + (y - 3)^2 = 36, then the others). Make sure the standard x- and y-axes are visible.

For one circle, visually inspect where it meets the x-axis (horizontal axis) and y-axis (vertical axis). You can click on intersection points or use the Desmos point tool to see their coordinates and count how many such points lie on the axes.

Repeat the counting process for each of the four circles, possibly hiding the others to avoid clutter. Look for the circle that touches one axis at exactly one point and the other axis at exactly two points, for a total of three distinct axis-intersections—that equation corresponds to the correct choice.

All choices are in the standard circle form

where the center is $(h,k)$ and the radius is $r$.

Identify $(h,k)$ and $r$ for each option:

• A) $(x + 4)^2 + (y - 3)^2 = 36$: center $(-4,3)$, radius $6$.
• B) $(x - 6)^2 + (y + 2)^2 = 9$: center $(6,-2)$, radius $3$.
• C) $(x - 3)^2 + (y - 4)^2 = 16$: center $(3,4)$, radius $4$.
• D) $(x + 1)^2 + (y - 7)^2 = 25$: center $(-1,7)$, radius $5$.

For a point $(h,k)$:

• Distance to the x-axis (the line $y=0$) is $|k|$.
• Distance to the y-axis (the line $x=0$) is $|h|$.

For a circle of radius $r$:

• If distance to an axis $<$ $r$: the circle crosses that axis in 2 points.
• If distance to an axis $=$ $r$: the circle is tangent to that axis (touches it once) → 1 point.
• If distance to an axis $>$ $r$: the circle does not reach that axis → 0 points.

We need a total of 3 distinct axis-intersections, so we are looking for a circle that has either:

• 1 intersection with one axis and 2 with the other (since $1 + 2 = 3$).

(In these choices, none of the circles pass through the origin, so there is no overlap between x- and y-axis intersection points.)

Use the distance–radius comparison for each equation.

Choice A: center $(-4,3)$, radius $6$.

• Distance to x-axis: $|3| = 3 < 6$ → 2 x-axis intersections.
• Distance to y-axis: $|-4| = 4 < 6$ → 2 y-axis intersections.
• Total: $2 + 2 = 4$ axis-intersections.

Choice B: center $(6,-2)$, radius $3$.

• Distance to x-axis: $|-2| = 2 < 3$ → 2 x-axis intersections.
• Distance to y-axis: $|6| = 6 > 3$ → 0 y-axis intersections.
• Total: $2 + 0 = 2$ axis-intersections.

Choice C: center $(3,4)$, radius $4$.

• Distance to x-axis: $|4| = 4$.
• Distance to y-axis: $|3| = 3$.

Choice D: center $(-1,7)$, radius $5$.

• Distance to x-axis: $|7| = 7 > 5$ → 0 x-axis intersections.
• Distance to y-axis: $|{-1}| = 1 < 5$ → 2 y-axis intersections.
• Total: $0 + 2 = 2$ axis-intersections.

Only one choice will end up with a total of exactly 3 intersections once you interpret the distances for that circle.

Now finish the analysis for the remaining circle from Step 3.

For Choice C, center $(3,4)$, radius $4$:

• Distance to x-axis: $|4| = 4 = r$ → the circle is tangent to the x-axis → 1 intersection with the x-axis.
• Distance to y-axis: $|3| = 3 < 4$ → the circle crosses the y-axis → 2 intersections with the y-axis.

Total intersections with the coordinate axes: $1 + 2 = 3$ distinct points.

Therefore, the equation that represents a circle intersecting the coordinate axes at exactly three distinct points is