Contains the points (-2,0 and (8,0), radius 5

To find the equation of a circle that passes through the points \((-2, 0)\) and \((8, 0)\) and has a radius of 5, we need to follow these steps:

### 1. **Determine the center of the circle**:
   - Since both points lie on the circle and on the x-axis (they are of the form \((x, 0)\)), the center must lie on the x-axis as well.
   - Let the center of the circle be \((h, k)\). Since it lies on the x-axis, \(k = 0\). Thus, the center is \((h, 0)\).

### 2. **Use the distance formula to find \(h\)**:
   - The radius \(r = 5\).
   - The distance between the center \((h, 0)\) and each point on the circle must be equal to the radius. Let's set up the equation using one of the points, say \((-2, 0)\):

\[
\sqrt{(h - (-2))^2 + (0 - 0)^2} = 5
\]

\[
\sqrt{(h + 2)^2} = 5
\]

\[
|h + 2| = 5
\]

This gives us two possible solutions for \(h\):

1. \(h + 2 = 5 \implies h = 3\)
2. \(h + 2 = -5 \implies h = -7\)

Thus, the two possible centers of the circle are \((3, 0)\) and \((-7, 0)\).

### 3. **Write the equation of the circle**:
   - The general equation of a circle is \((x - h)^2 + (y - k)^2 = r^2\).
   - Since \(k = 0\) and \(r = 5\), the equation simplifies to:

For \(h = 3\):
\[
(x - 3)^2 + y^2 = 25
\]

For \(h = -7\):
\[
(x + 7)^2 + y^2 = 25
\]

### **Final Answer**:
- The equations of the circles that satisfy the conditions are:
  1. \((x - 3)^2 + y^2 = 25\)
  2. \((x + 7)^2 + y^2 = 25\)

These represent the two possible circles.

Would you like further details on these solutions or have any questions?

### Related Questions:
1. How do you determine the distance between two points on a plane?
2. What are the different forms of the equation of a circle?
3. How do you solve absolute value equations like \(|x + 2| = 5\)?
4. Can a circle have more than one equation based on the same conditions?
5. What happens to the circle's equation if the radius changes?
6. How do you find the midpoint between two points?
7. How can you check if a point lies on a given circle?
8. What is the significance of the x-intercepts in the context of a circle?

**Tip**: When solving geometric problems, it's helpful to sketch the scenario to visualize the relationships between points, lines, and curves.

Learn how to find the equation of a circle passing through the points (-2, 0) and (8, 0) with a radius of 5. This problem involves using the distance formula and understanding circle equations. Suitable for advanced high school students.

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