Final answer:
The given equation x² + y² - 10x - 10y + 25 = 0, after completing the square for x and y, is rewritten in the form (x - 5)² + (y - 5)² = 25, which is the standard form of a circle.
Step-by-step explanation:
To determine whether the equation x² + y² - 10x - 10y + 25 = 0 represents a circle, ellipse, parabola, or hyperbola, we can complete the square for both x and y terms.
Rearrange the equation:
(x² - 10x) + (y² - 10y) + 25 = 0
Add and subtract the necessary terms to complete the square for x and y:
(x² - 10x + 25) - 25 + (y² - 10y + 25) - 25 + 25 = 0
(x - 5)² + (y - 5)² = 25
This is the standard form for a circle with the center at (5, 5) and a radius of 5.