Answer:
To find the center of the given equation of a circle, we first need to rewrite it in the standard form:
x² + y² + 8x = 2x - 20y - 105
x² + 8x + y² = 2x - 20y - 105
Completing the square for the x terms:
(x² + 8x + 16) + y² = 2x - 20y - 105 + 16
(x + 4)² + y² = 2x - 20y - 89
Now we can see that the equation is in the standard form:
(x - (-4))² + (y - 0)² = r²
where the center of the circle is (-4, 0) and the radius squared is:
r² = 2x - 20y - 89
Note that the center of the circle can be found by taking the negative of the coefficients of x and y in the completed square terms and reversing their signs. In this case, the completed square term for x is (x + 4)², so the x-coordinate of the center is -4. The completed square term for y is y², so the y-coordinate of the center is 0.