Final answer:
To graph the given circle equation, we complete the square for both x and y terms. The equation simplifies to the standard form of a circle with center at (5, 3) and radius 4, which is option A.
Step-by-step explanation:
To graph the circle described by the equation x² + y² - 10x - 6y + 18 = 0, we first need to complete the square for the x and y terms to get the equation into the standard form of a circle's equation, which is (x - h)² + (y - k)² = r², where (h, k) is the center of the circle and r is the radius.
Completing the square for the x terms:
- x² - 10x can be rewritten as (x - 5)², since (x - 5)(x - 5) = x² - 10x + 25. We then subtract 25 to keep the equation balanced.
Completing the square for the y terms:
- y² - 6y can be rewritten as (y - 3)², since (y - 3)(y - 3) = y² - 6y + 9. We then subtract 9 to keep the equation balanced.
So the equation becomes:
(x - 5)² + (y - 3)² - 25 - 9 + 18 = 0
Which simplifies to:
(x - 5)² + (y - 3)² - 16 = 0
Then,
(x - 5)² + (y - 3)² = 16
From this we can tell that the center of the circle is (5, 3) and the radius is 4 since r² = 16 implies r = 4.
The correct graph is Circle with center (5, 3) and radius 4, which corresponds to option A.