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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.

A circle has a radius of √13 units and passes through point A(3,1). Find its equation if its center lies on the y-axis. i.e., the center is at (0, k).

Learn how to find the equation of a circle with a radius of √13 that passes through point A(3,1) and has its center on the y-axis. This detailed solution covers core concepts from geometry and coordinate geometry, ideal for students in Grades 9-12.

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