Final answer:
After simplification, we find that the center is (-6, 5) and the radius is 7.
Step-by-step explanation:
The equation of a circle is given as x^2+12x+y^2-10y+12=0. To identify the center and radius, we need to complete the square for both x and y terms.
- Rewrite the equation grouping the x terms and y terms:
(x^2 + 12x) + (y^2 - 10y) = -12.
- Complete the square for the x terms:
Add (12/2)^2 = 36 to both sides to balance the equation.
- Complete the square for the y terms:
Add (10/2)^2 = 25 to both sides to balance the equation.
- Now the equation looks like this:
(x^2 + 12x + 36) + (y^2 - 10y + 25) = 49.
- Factor the perfect squares:
(x + 6)^2 + (y - 5)^2 = 7^2.
This is now in the standard form of a circle equation (x - h)^2 + (y - k)^2 = r^2, where (h,k) is the center and r is the radius.
Therefore, the center of the circle is (-6, 5) and the radius is 7.