To find the radius of a circle from its equation, we can use the standard form of a circle’s equation:
(x – h)² + (y – k)² = r²
Here, (h, k) represents the center of the circle, and r is the radius. The first step is to rewrite the given equation in the standard form.
The provided equation is:
x² + 2x + y² + 8y + 16 = 0
Next, we rearrange it to isolate the constant term:
x² + 2x + y² + 8y = -16
Now, we complete the square for the x and y terms separately. For the x terms (x² + 2x):
1. Take the coefficient of x, which is 2. Divide it by 2 to get 1, and then square it to get 1.
2. Add and subtract this value inside the equation:
x² + 2x + 1 – 1
This simplifies to:
(x + 1)² – 1
For the y terms (y² + 8y):
1. Take the coefficient of y, which is 8. Divide by 2 to get 4, and then square it to get 16.
2. Add and subtract this value:
y² + 8y + 16 – 16
This gives:
(y + 4)² – 16
Substituting these back into our equation gives:
((x + 1)² – 1) + ((y + 4)² – 16) = -16
Now simplify:
(x + 1)² + (y + 4)² – 17 = -16
Which leads to:
(x + 1)² + (y + 4)² = 1
Now, we can see that the circle is centered at (-1, -4) and the radius squared is 1.
Finally, to find the radius, we take the square root:
r = √1 = 1
Therefore, the radius of the circle whose equation is given is 1 unit.