Final answer:
The equation x²+y²-12x-14y+21=0 represents a circle with the center at (6, 7) and a radius of 8 after completing the square for x and y.
Step-by-step explanation:
To find the center and radius of the circle given by the equation x²+y²-12x-14y+21=0, we need to complete the square for both x and y.
First, we rearrange the terms to group x's and y's together:
x²-12x+y²-14y = -21
Next, we complete the square for both x and y by adding (12/2)² = 36 to both sides for x and (14/2)² = 49 for y:
x²-12x+36+y²-14y+49 = -21+36+49
Now, we can write the equation as two perfect squares:
(x-6)²+(y-7)² = 64
The equation now represents a circle with the center at (6,7) and a radius of 8, since 64 is the square of 8.
The circle's center is (6, 7) and its radius is 8.