give the standard equation of the circle satisfying the given condition.

Given the points \((-2,0)\) and \( (8,0)\) on the circumference and the radius \(5\), we have determined the possible centers of the circle to be \((3, 5)\) and \((3, -5)\). 

The standard equation of a circle with center \((h, k)\) and radius \(r\) is:
\[
(x - h)^2 + (y - k)^2 = r^2
\]

Using the first possible center \((3, 5)\) and radius \(5\):
\[
(x - 3)^2 + (y - 5)^2 = 25
\]

Using the second possible center \((3, -5)\) and radius \(5\):
\[
(x - 3)^2 + (y + -5)^2 = 25
\]

So, the standard equations of the circles are:
1. \((x - 3)^2 + (y - 5)^2 = 25\)
2. \((x - 3)^2 + (y + 5)^2 = 25\)

Would you like more details or have any questions about these equations?

Here are some related questions for further practice:

1. How would you derive the equation of the circle if the center and radius are given directly?
2. How can you verify if a given point lies on the circle using the standard equation?
3. What transformations would shift the circle’s center from \((3, 5)\) to another point?
4. How would you find the equation of the tangent line to the circle at a given point on its circumference?
5. How can you determine the length of the chord connecting two points on the circle?
6. What is the general form of the equation of a circle and how does it relate to the standard form?
7. How do the equations change if the radius of the circle increases or decreases?
8. What happens to the circle's equation if it is translated horizontally or vertically?

**Tip:** When writing the standard equation of a circle, always double-check the values for the center and the radius to ensure accuracy.

Explore the standard equation of a circle given two points on its circumference and the radius. Learn how to derive the equation, understand the concept of circle center and radius, and solve related problems. Suitable for high school students and geometry enthusiasts.

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