Final answer:
To find the radius of the circle described by the equation x²-2x+y²-6y=6, we must complete the square to put it in standard form and then compare it with the equation of a circle. The completed equation is (x-1)² + (y-3)² = 16, hence the radius of the circle is 4.
Step-by-step explanation:
The equation given represents a circle in the xy-plane. To find the radius of this circle, we must first complete the square for both x and y terms. The original equation is x²-2x+y²-6y=6. Completing the square for the x terms, we add and subtract (2/2)² = 1 inside the equation, resulting in x²-2x+1 - 1. Similarly, we do this for the y terms by adding and subtracting (6/2)² = 9, resulting in y²-6y+9 - 9. This gives us:
(x²-2x+1) + (y²-6y+9) = 6 + 1 + 9
Which simplifies to:
(x-1)² + (y-3)² = 16
The equation of a circle in standard form is (x-h)² + (y-k)² = r², where (h,k) is the center of the circle and r is the radius. Comparing our equation to this standard form, we see that the radius squared, r², is equal to 16. Therefore, the radius of the circle is √16, which equals 4.