In the standard form (x+2)^2+(y-7)^2=8 what is the center and radius?

The given equation \((x+2)^2 + (y-7)^2 = 8\) is in the standard form of a circle, which is:

\[
(x - h)^2 + (y - k)^2 = r^2
\]

Here:
- \((h, k)\) is the center of the circle.
- \(r\) is the radius of the circle.

To find the center and radius:
1. **Center**: Compare the given equation to the standard form.
   - \((x - (-2))^2 + (y - 7)^2 = 8\)
   - Thus, \(h = -2\) and \(k = 7\).
   - So, the center is \((-2, 7)\).

2. **Radius**: The right-hand side of the equation is \(r^2 = 8\).
   - Take the square root of 8 to find the radius: \(r = \sqrt{8} = 2\sqrt{2}\).

**Answer**:
- **Center**: \((-2, 7)\)
- **Radius**: \(2\sqrt{2}\)

Would you like more details on this solution or have any other questions?

Here are some related questions you might find useful:
1. How do you convert a general form equation of a circle to the standard form?
2. What are the steps to derive the equation of a circle from its center and radius?
3. How can you determine the equation of a circle given three points on its circumference?
4. How do you find the distance between the centers of two circles?
5. How does the equation of a circle change when it is translated on the coordinate plane?
6. What is the significance of completing the square in finding the center and radius of a circle?
7. How can you determine if a given point lies inside, on, or outside a circle?
8. How do you find the equation of a circle tangent to a line?

**Tip**: Remember that the radius is always positive, so when extracting the square root, ensure you take the positive root for the radius.

Learn how to find the center and radius of a circle given its equation (x+2)^2 + (y-7)^2 = 8. This problem covers coordinate geometry concepts, including the standard form of a circle equation and methods to determine center and radius. Suitable for students in Grades 9-12.

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