Final answer:
The center of the circle given by the equation x² + y² - 8x – 12y - 8 = 0 is (4, 6) after completing the square, and the radius is 2√15.
Step-by-step explanation:
To find the center and radius of the circle given by the equation x² + y² - 8x – 12y - 8 = 0, we need to first complete the square for both the x and y terms. This process involves creating perfect square trinomials from the given equation.
We rewrite the equation by grouping the x and y terms and moving the constant to the other side:
x² - 8x + y² - 12y = 8
Now we complete the square for the x and y groups (finding the number that makes x and y into perfect squares):
(x - 4)² +(y - 6)² = 8 + 16 + 36
After adding 16 and 36 to both sides to complete the squares, the equation becomes:
(x - 4)² +(y - 6)² = 60
The equation now has the standard form of a circle (x - h)² + (y - k)² = r², where (h, k) is the center of the circle and r is the radius. So the center of the circle is (4, 6) and the radius is the square root of 60, which simplifies to 2√15.