Finding the Center and Radius of the Circle
To find the center and radius of the circle given the equation x² + 2x + y² + 4y = 20, we need to rewrite this equation in the standard form of a circle, which is:
(x – h)² + (y – k)² = r²
where (h, k) is the center of the circle and r is the radius.
Step 1: Rearrange the Equation
First, let’s group the x terms and the y terms together:
x² + 2x + y² + 4y = 20
Step 2: Complete the Square
Next, we complete the square for the x terms and the y terms.
For the x terms:
The expression is x² + 2x. To complete the square:
- Take half of the coefficient of x (which is 2), square it: (2/2)² = 1.
- Add and subtract this value inside the equation:
x² + 2x + 1 - 1 + y² + 4y = 20
For the y terms:
The expression is y² + 4y. To complete the square:
- Take half of the coefficient of y (which is 4), square it: (4/2)² = 4.
- Add and subtract this value:
x² + 2x + 1 + y² + 4y + 4 - 1 - 4 = 20
Step 3: Simplifying the Equation
Now, rewrite the equation:
(x + 1)² + (y + 2)² = 25
Here, we combine the constants on the right side:
20 + 1 + 4 = 25
Step 4: Identify the Center and Radius
Now, we can identify the center (h, k) and the radius r:
- From the equation (x + 1)² + (y + 2)² = 25:
- Center, (h, k) = (-1, -2)
- Radius, r = √25 = 5
Conclusion
The center of the circle is at the point (-1, -2), and the radius is 5.