Definition: The equation of a circle with center (h, k) and radius r is (x – h)^2 + (y – k)^2 = r^2, which comes from the distance formula.
Where does it come from?
Every point on a circle is at distance r from the center. Writing that distance with the distance formula and squaring both sides gives the equation.
How do you read it?
The center is (h, k), with signs opposite to those shown. The number on the right is r2, so take its square root to find the radius.
Example. (x + 4)2 + (y – 7)2 = 50 has center (-4, 7) and radius the square root of 50, about 7.07. To go the other way, a circle with center (3, -2) passing through (8, 10) has r2 = 25 + 144 = 169, so its equation is (x – 3)2 + (y + 2)2 = 169.
How do you handle expanded form?
If the equation is x2 + y2 – 6x + 10y – 2 = 0, complete the square in x and in y. That gives (x – 3)2 + (y + 5)2 = 36, so the center is (3, -5) and r = 6. Common errors: reading the center with the wrong signs, and reporting r2 as the radius.