To find the center and radius of the given equation, we first need to rewrite it in the standard form of a circle’s equation, which is: (x – h)² + (y – k)² = r², where (h, k) is the center and r is the radius.
The given equation is:
x² + 2x + y² – 4y = 20
First, we will rearrange the equation by completing the square for both x and y terms.
Step 1: Completing the square for x
Take the x terms:
x² + 2x
To complete the square, take half of the coefficient of x (which is 2), square it (1), and adjust the equation:
x² + 2x + 1 – 1
This can be rewritten as:
(x + 1)² – 1
Step 2: Completing the square for y
Now take the y terms:
y² – 4y
Again, take half of -4, square it (which is 4), and adjust the equation:
y² – 4y + 4 – 4
This can be rewritten as:
(y – 2)² – 4
Step 3: Substitute back into the equation
Now we substitute everything back into the original equation:
(x + 1)² – 1 + (y – 2)² – 4 = 20
Combine the constants on the left side:
(x + 1)² + (y – 2)² – 5 = 20
So:
(x + 1)² + (y – 2)² = 25
Step 4: Identify the center and radius
Now we can see that this is in the standard form:
(x – (-1))² + (y – 2)² = 5²
The center of the circle is (-1, 2) and the radius is 5.
Conclusion
The center is located at (-1, 2) and the radius is 5.