Final answer:
The equation 81x² + 81y² - 72x + 72y - 32 = 0 represents a circle. The center of the circle is (4/9, -4/9) and the radius is 8/9.
Step-by-step explanation:
The equation 81x² + 81y² - 72x + 72y - 32 = 0 represents a circle. To determine the center and radius of the circle, we need to rewrite the equation in the standard form: (x - h)² + (y - k)² = r², where (h, k) is the center of the circle and r is the radius.
Expanding the equation, we get 81(x² - 8/9x) + 81(y² + 8/9y) = 32. Completing the square, we have 81(x² - 8/9x + 4/9²) + 81(y² + 8/9y + 4/9²) = 32 + 81(4/9) + 81(4/9).
Therefore, the equation can be rewritten as (x - 4/9)² + (y + 4/9)² = 32/81 + 16/81 + 16/81. Hence, the center of the circle is (4/9, -4/9) and the radius is √(32/81 + 16/81 + 16/81), which simplifies to √(64/81), or 8/9.