https://file.aiquickdraw.com/mathbot/file/1729202512587y8ou69gg.jpg

To determine the center and radius of the circle given by the equation:

\[
x^2 + y^2 - 2x + 4y - 20 = 0
\]

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 of the circle and \(r\) is the radius.

### Steps:
1. **Group the \(x\) and \(y\) terms**:
   \[
   (x^2 - 2x) + (y^2 + 4y) = 20
   \]

2. **Complete the square**:
   - For the \(x\)-terms, take half of \(-2\), square it, and add it to both sides:
     \[
     (x^2 - 2x) \quad \text{becomes} \quad (x - 1)^2 - 1
     \]
   - For the \(y\)-terms, take half of \(4\), square it, and add it to both sides:
     \[
     (y^2 + 4y) \quad \text{becomes} \quad (y + 2)^2 - 4
     \]

3. **Substitute the completed squares** into the equation:
   \[
   (x - 1)^2 - 1 + (y + 2)^2 - 4 = 20
   \]

4. **Simplify the constants** on the right-hand side:
   \[
   (x - 1)^2 + (y + 2)^2 = 20 + 1 + 4
   \]
   \[
   (x - 1)^2 + (y + 2)^2 = 25
   \]

Now the equation is in standard form:
\[
(x - 1)^2 + (y + 2)^2 = 25
\]

### Conclusion:
- The center of the circle is \((1, -2)\).
- The radius is \(\sqrt{25} = 5\).

Would you like further clarification or additional details?

### Related Questions:
1. What is the general method for completing the square?
2. How can we identify the center and radius directly from the standard form of a circle equation?
3. How do you solve for the equation of a circle given the center and a point on the circle?
4. How would you graph this circle on a coordinate plane?
5. What is the geometric significance of completing the square?

### Tip:
When completing the square, always remember to add the same constants to both sides of the equation to maintain equality.

Determine the coordinates for the center and radius of the following circle: x² + y² - 2x + 4y - 20 = 0

Find the center and radius of the circle given by the equation x² + y² - 2x + 4y - 20 = 0. Learn the step-by-step process, including completing the square, to rewrite the equation in standard form.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation