Answer:
![(x + 5) ^ 2 + (y-0) ^ 2 = 2 ^ 2](https://tex.z-dn.net/?f=%28x%20%2B%205%29%20%5E%202%20%2B%20%28y-0%29%20%5E%202%20%3D%202%20%5E%202)
Step-by-step explanation:
The standard form for the equation of a circumference is:
![(x-a) ^ 2 + (y-b) ^ 2 = r ^ 2](https://tex.z-dn.net/?f=%28x-a%29%20%5E%202%20%2B%20%28y-b%29%20%5E%202%20%3D%20r%20%5E%202)
Where:
(a, b) is the center of the circumference
r is the radius
In this problem we have the equation of the following circumference, and we want to convert it to the standard form:
![x ^ 2 + y ^ 2 + 10x +21 = 0](https://tex.z-dn.net/?f=x%20%5E%202%20%2B%20y%20%5E%202%20%2B%2010x%20%2B21%20%3D%200)
The first thing we must do to transform this equation into the standard form is to use the square completion technique.
The steps are shown below:
1. Group all the same variables:
![y ^ 2 + (x ^ 2 + 10x) = -21](https://tex.z-dn.net/?f=y%20%5E%202%20%2B%20%28x%20%5E%202%20%2B%2010x%29%20%3D%20-21)
2. Take the coefficient that accompanies the variable x. In this case the coefficient is 10. Then, divide by 2 and the result elevate it to the square.
We have:
![\frac{10}{2} = 5\\\\(\frac{10}{2}) ^ 2 = 25](https://tex.z-dn.net/?f=%5Cfrac%7B10%7D%7B2%7D%20%3D%205%5C%5C%5C%5C%28%5Cfrac%7B10%7D%7B2%7D%29%20%5E%202%20%3D%2025)
3. Add on both sides of the equality the term obtained in the previous step:
![y ^ 2 + (x ^ 2 + 10x +25) = -21 +25\\\\y ^ 2 + (x ^ 2 + 10x +25) = 4](https://tex.z-dn.net/?f=y%20%5E%202%20%2B%20%28x%20%5E%202%20%2B%2010x%20%2B25%29%20%3D%20-21%20%2B25%5C%5C%5C%5Cy%20%5E%202%20%2B%20%28x%20%5E%202%20%2B%2010x%20%2B25%29%20%3D%204)
4. Factor the resulting expression, and you will get:
![(x + 5) ^ 2 + y ^ 2 = 4](https://tex.z-dn.net/?f=%28x%20%2B%205%29%20%5E%202%20%2B%20y%20%5E%202%20%3D%204)
Write the equation in the standard form:
![(x + 5) ^ 2 + (y-0) ^ 2 = 2 ^ 2](https://tex.z-dn.net/?f=%28x%20%2B%205%29%20%5E%202%20%2B%20%28y-0%29%20%5E%202%20%3D%202%20%5E%202)
Then, the center is the point (5, 0) and the radius is r = 2.
Observe the attached image