find the center and radius of each circle
x²+y²-4x-16y+32=0

To find the center and radius of the circle given by the equation \( x^2 + y^2 - 4x - 16y + 32 = 0 \), we can rewrite the equation in standard form by completing the square for both \( x \) and \( y \).

### Step 1: Rewrite the equation grouping the \( x \) and \( y \) terms:
\[
x^2 - 4x + y^2 - 16y = -32
\]

### Step 2: Complete the square for the \( x \) terms:
- Take the coefficient of \( x \), which is \(-4\), divide by 2, and square it: 
\[
\left(\frac{-4}{2}\right)^2 = 4
\]
- Add and subtract 4 inside the equation:
\[
x^2 - 4x + 4
\]

### Step 3: Complete the square for the \( y \) terms:
- Take the coefficient of \( y \), which is \(-16\), divide by 2, and square it:
\[
\left(\frac{-16}{2}\right)^2 = 64
\]
- Add and subtract 64 inside the equation:
\[
y^2 - 16y + 64
\]

### Step 4: Rewrite the equation incorporating the squares:
\[
(x^2 - 4x + 4) + (y^2 - 16y + 64) = -32 + 4 + 64
\]
\[
(x - 2)^2 + (y - 8)^2 = 36
\]

### Step 5: Identify the center and radius:
The standard form of a circle's equation is:
\[
(x - h)^2 + (y - k)^2 = r^2
\]
Where \((h, k)\) is the center, and \(r\) is the radius.

- Center: \((h, k) = (2, 8)\)
- Radius: \(r = \sqrt{36} = 6\)

### Final Answer:
- **Center**: \((2, 8)\)
- **Radius**: \(6\)

Would you like more details or have any questions about the process? Here are some related questions:

1. How do you complete the square for a quadratic expression?
2. Can you find the equation of a circle with a different center and radius?
3. What is the geometric interpretation of the center and radius of a circle?
4. How do changes in the equation coefficients affect the circle’s position and size?
5. How can you derive the distance formula using the circle’s equation?

**Tip:** Always check your work by expanding the standard form equation back to its original form to ensure accuracy.

Discover how to find the center and radius of the circle defined by the equation x^2 + y^2 - 4x - 16y + 32 = 0. This problem involves completing the square for circle equations and understanding the standard form. Suitable for Grades 10-12 students.

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