If a circle’s center is on the x-axis, what is its y-coordinate? If it is tangent to the y-axis (the line $x = 0$), how is the radius related to the x-coordinate of the center?

For two circles to be tangent (touch at exactly one point), how should the distance between their centers relate to their radii? Consider both external tangency (touching from the outside) and internal tangency (one inside the other).

For each choice, find the center and radius, then compute the distance between its center and the center of circle C. Compare that distance with the sum and the difference of the two radii to see if they are tangent.

In Desmos, enter the equation (x-3)^2 + y^2 = 25 to see circle C. Note its center at (3, 0) and how far it extends left and right.

One at a time, type each option into Desmos: (x-1)^2 + y^2 = 1, (x-4)^2 + y^2 = 16, (x-5)^2 + y^2 = 25, (x-8)^2 + y^2 = 64. For each, check that the circle just touches the y-axis at exactly one point (tangency to the y-axis).

With all circles visible, look at the intersections between circle C and each candidate circle D. The correct choice will be the one whose circle touches circle C at exactly one point (a single point of intersection), while the others will either not meet or intersect in two points.

Circle C is given by $(x - 3)^{2} + y^{2} = 25$.

• This is in the standard form $(x - h)^{2} + (y - k)^{2} = r^{2}$.
• So the center of circle C is $(3, 0)$ and the radius is $r = 5$ (since $25 = 5^{2}$).

Circle D:

• Has its center on the x-axis, so its center is $(a, 0)$ for some $a$.
• Is tangent to the y-axis (the line $x = 0$). The distance from $(a, 0)$ to the y-axis is $|a|$, and this distance must equal the radius of circle D.

So for circle D, if its center is $(a, 0)$ and its radius is $r$, then $r = |a|$.

Check each answer choice:

• A) $(x - 1)^{2} + y^{2} = 1$: center $(1, 0)$, radius $1$.
• B) $(x - 4)^{2} + y^{2} = 16$: center $(4, 0)$, radius $4$.
• C) $(x - 5)^{2} + y^{2} = 25$: center $(5, 0)$, radius $5$.
• D) $(x - 8)^{2} + y^{2} = 64$: center $(8, 0)$, radius $8$.

All four circles have center on the x-axis and radius equal to the x-coordinate, so all are tangent to the y-axis. The remaining condition to use is tangency to circle C.

For two circles with centers $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ and radii $r_{1}$ and $r_{2}$:

• If they are tangent externally (touch from the outside), the distance between centers equals $r_{1} + r_{2}$.
• If they are tangent internally (one inside the other, touching at one point), the distance between centers equals $|r_{1} - r_{2}|$.

Circle C has center $(3, 0)$ and radius $5$. For each option, let circle D have center $(a, 0)$ and radius $r$.

• The distance between centers is $|a - 3|$ because both centers lie on the x-axis.
• For tangency, we need $|a - 3| = r + 5$ (external) or $|a - 3| = |5 - r|$ (internal).

You can now plug in $a$ and $r$ from each choice to see which one satisfies a tangency equation.

Test each circle D:

• A) Center $(1, 0)$, radius $1$:

  • Distance between centers: $|1 - 3| = 2$.
  • $r + 5 = 1 + 5 = 6$ and $|5 - r| = |5 - 1| = 4$.
  • Distance $2$ is not $6$ or $4$ → not tangent.

• B) Center $(4, 0)$, radius $4$:

  • Distance between centers: $|4 - 3| = 1$.
  • $r + 5 = 4 + 5 = 9$ and $|5 - r| = |5 - 4| = 1$.
  • Distance $1$ equals $|5 - r|$, so the circles are internally tangent (touch at exactly one point).

• C) Center $(5, 0)$, radius $5$:

  • Distance between centers: $|5 - 3| = 2$.
  • $r + 5 = 5 + 5 = 10$ and $|5 - r| = |5 - 5| = 0$.
  • Distance $2$ is not $10$ or $0$ → not tangent.

• D) Center $(8, 0)$, radius $8$:

  • Distance between centers: $|8 - 3| = 5$.
  • $r + 5 = 8 + 5 = 13$ and $|5 - r| = |5 - 8| = 3$.
  • Distance $5$ is not $13$ or $3$ → not tangent.

Only choice B makes the distance between centers equal to the difference of the radii, so circle D is tangent to circle C at exactly one point. Thus the equation of circle D is $(x - 4)^{2} + y^{2} = 16$.