Answer:
[tex](x + 5) ^ 2 + (y-0) ^ 2 = 2 ^ 2[/tex]
Step-by-step explanation:
The standard form for the equation of a circumference is:
[tex](x-a) ^ 2 + (y-b) ^ 2 = r ^ 2[/tex]
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:
[tex]x ^ 2 + y ^ 2 + 10x +21 = 0[/tex]
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:
[tex]y ^ 2 + (x ^ 2 + 10x) = -21[/tex]
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:
[tex]\frac{10}{2} = 5\\\\(\frac{10}{2}) ^ 2 = 25[/tex]
3. Add on both sides of the equality the term obtained in the previous step:
[tex]y ^ 2 + (x ^ 2 + 10x +25) = -21 +25\\\\y ^ 2 + (x ^ 2 + 10x +25) = 4[/tex]
4. Factor the resulting expression, and you will get:
[tex](x + 5) ^ 2 + y ^ 2 = 4[/tex]
Write the equation in the standard form:
[tex](x + 5) ^ 2 + (y-0) ^ 2 = 2 ^ 2[/tex]
Then, the center is the point (5, 0) and the radius is r = 2.
Observe the attached image
