To convert the given equation into standard form, we need to rearrange it and complete the square for both the x and y terms.
The equation is:
x² + y² + 8x + 22y + 37 = 0
First, let’s move the constant term to the other side:
x² + y² + 8x + 22y = -37
Next, let’s group the x terms and the y terms:
(x² + 8x) + (y² + 22y) = -37
Now, we need to complete the square for the x terms and the y terms.
For the x terms, x² + 8x:
- Take half of 8, which is 4, and square it to get 16.
- Add and subtract 16:
(x² + 8x + 16 – 16) = (x + 4)² – 16
For the y terms, y² + 22y:
- Take half of 22, which is 11, and square it to get 121.
- Add and subtract 121:
(y² + 22y + 121 – 121) = (y + 11)² – 121
Putting it all back together, we can rewrite our equation:
(x + 4)² – 16 + (y + 11)² – 121 = -37
Now simplify this:
(x + 4)² + (y + 11)² – 137 = -37
Adding 137 to both sides gives us:
(x + 4)² + (y + 11)² = 100
Now, the equation is in standard form:
(x + 4)² + (y + 11)² = 10²
This represents a circle with center at (-4, -11) and a radius of 10.