Final answer:
To write the equation in standard form, we complete the square for the y terms. The resulting standard form of the circle's equation is x² + (y - 3)² = 40, with the center at (0, 3) and the radius as √40.
Step-by-step explanation:
To write the equation y2 - 6y = -x2 + 31 in standard form for a circle, we need to complete the square for both the x and y terms.
Starting with the given equation:
y2 - 6y = -x2 + 31
First, let's move the x2 term to the left side:
x2 + y2 - 6y = 31
Now, we complete the square for the y terms. To do this, we take half of the coefficient of the y term (which is -6), square it (giving us 9), and add it to both sides:
x2 + y2 - 6y + 9 = 31 + 9
This simplifies to:
x2 + (y - 3)2 = 40
Now we have the equation of the circle in standard form, which is (x - h)2 + (y - k)2 = r2, where (h,k) is the center of the circle and r is the radius. Comparing with the standard form, the center of the circle is (h, k) = (0, 3) and the radius squared is 40, so the radius r is √40.
The standard form of the given equation is:
x2 + (y - 3)2 = 40