We are given a circle with:

• Radius $r = \sqrt{13}$,
• The circle passes through the point $A(3,1)$,
• The center of the circle is on the y-axis, at $(0, k)$.

General form of the equation of a circle:

$(x - h)^2 + (y - k)^2 = r^2$ Where:

• $(h, k)$ is the center of the circle,
• $r$ is the radius.

Since the center is on the y-axis, $h = 0$, so the equation simplifies to: $x^2 + (y - k)^2 = r^2$

Now, substituting the given radius $r = \sqrt{13}$ into the equation: $x^2 + (y - k)^2 = 13$

Using point $A(3,1)$ to find $k$:

Point $A(3, 1)$ lies on the circle, so we can substitute $x = 3$ and $y = 1$ into the circle equation: $3^2 + (1 - k)^2 = 13$ $9 + (1 - k)^2 = 13$ $(1 - k)^2 = 13 - 9 = 4$ $1 - k = \pm 2$ This gives two possible values for $k$: $k = 1 - 2 = -1 \quad \text{or} \quad k = 1 + 2 = 3$

The two possible equations of the circle:

1. If $k = -1$: $x^2 + (y + 1)^2 = 13$
2. If $k = 3$: $x^2 + (y - 3)^2 = 13$

Thus, there are two possible equations for the circle:

1. $x^2 + (y + 1)^2 = 13$
2. $x^2 + (y - 3)^2 = 13$

Let me know if you'd like further details or clarification on this solution!

Here are five follow-up questions for you:

1. What is the general equation of a circle not necessarily centered on the y-axis?
2. How can you use the distance formula to verify the radius of a circle given two points?
3. What is the geometric significance of having two possible values for $k$?
4. How do you determine if a point lies inside, on, or outside of a circle?
5. How would the equation change if the center was not constrained to the y-axis?

Tip: When solving for the center of a circle, using points that lie on the circle and substituting them into the equation is a powerful method.