Identify the center and radius of the circle with the given equation. Sketch the graph. 2. 4x ^ 2 + 4y ^ 2 + 40x - 32y = 5

To identify the center and radius of the circle given by the equation \(4x^2 + 4y^2 + 40x - 32y = 5\), we need to rewrite the equation in the standard form of a circle \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius.

### Step-by-Step Solution

1. **Divide the entire equation by 4** to simplify:
   \[
   x^2 + y^2 + 10x - 8y = \frac{5}{4}
   \]

2. **Rearrange the terms** to prepare for completing the square:
   \[
   x^2 + 10x + y^2 - 8y = \frac{5}{4}
   \]

3. **Complete the square** for the \(x\)-terms and \(y\)-terms:
   
   - For \(x\)-terms: \(x^2 + 10x\)
     - Take half of 10 (which is 5) and square it: \(5^2 = 25\)
     - Add and subtract 25 inside the equation:
       \[
       x^2 + 10x + 25 - 25
       \]
   
   - For \(y\)-terms: \(y^2 - 8y\)
     - Take half of -8 (which is -4) and square it: \((-4)^2 = 16\)
     - Add and subtract 16 inside the equation:
       \[
       y^2 - 8y + 16 - 16
       \]

4. **Insert the completed squares** into the equation:
   \[
   (x^2 + 10x + 25) - 25 + (y^2 - 8y + 16) - 16 = \frac{5}{4}
   \]

5. **Rewrite the completed squares**:
   \[
   (x + 5)^2 - 25 + (y - 4)^2 - 16 = \frac{5}{4}
   \]

6. **Combine constants** on the right side of the equation:
   \[
   (x + 5)^2 + (y - 4)^2 - 41 = \frac{5}{4}
   \]
   \[
   (x + 5)^2 + (y - 4)^2 = \frac{5}{4} + 41
   \]
   \[
   (x + 5)^2 + (y - 4)^2 = \frac{5}{4} + \frac{164}{4}
   \]
   \[
   (x + 5)^2 + (y - 4)^2 = \frac{169}{4}
   \]

7. **Simplify the right side**:
   \[
   (x + 5)^2 + (y - 4)^2 = \left(\frac{13}{2}\right)^2
   \]

### Identifying the Center and Radius

The equation \((x + 5)^2 + (y - 4)^2 = \left(\frac{13}{2}\right)^2\) is now in the standard form \((x - h)^2 + (y - k)^2 = r^2\).

- The center \((h, k)\) is \((-5, 4)\).
- The radius \(r\) is \(\frac{13}{2}\).

### Sketching the Graph

To sketch the graph of the circle:

1. **Plot the center** at \((-5, 4)\).
2. **Draw the circle** with radius \(\frac{13}{2} = 6.5\).

Here's a simple visual representation of the circle:

![Circle](https://i.imgur.com/T4F6xOo.png)

### Summary

- **Center**: \((-5, 4)\)
- **Radius**: \(\frac{13}{2}\) or 6.5

Would you like more details or have any questions about this solution? Here are some related questions you might find helpful:

1. How do you complete the square for a quadratic expression?
2. What is the standard form of a circle's equation?
3. How do you derive the distance formula from the circle equation?
4. What is the geometric interpretation of completing the square?
5. How can you convert a general quadratic equation to the standard form of a circle?
6. What are the properties of a circle in the coordinate plane?
7. How do you determine the equation of a circle given its center and radius?
8. How does the equation of a circle change if it is translated in the coordinate plane?

**Tip:** When completing the square, always remember to add and subtract the same value inside the equation to maintain equality.

Learn how to identify the center and radius of a circle with the equation 4x^2 + 4y^2 + 40x - 32y = 5. Step-by-step solution including completing the square and graphing the circle. Suitable for high school students.

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