Notes worksheet on the equation of a circle, including how to determine if a point lies on a circle and how to write the equation given the center and a point.

- To determine if a point is on a circle, substitute the point into the formula for x and y.
- Example: Is (2, 3) on the circle (x-2)² + (y+1)² = 16?
- Substitute x=2, y=3: (2-2)² + (3+1)² = 0 + 16 = 16. Yes, it is on the circle.
- Example: Is (0, 5) on the circle (x+1)² + (y-4)² = 1?
- Substitute x=0, y=5: (0+1)² + (5-4)² = 1 + 1 = 2 ≠ 1. No, it is not on the circle.
- Try: Is (5, 2) on the circle (x+1)² + (y-2)² = 36?
- Substitute x=5, y=2: (5+1)² + (2-2)² = 36 + 0 = 36. Yes, it is on the circle.
- Writing the Equation Given the Center and a Point
1. Substitute the center and point into the formula for h, k, x, and y.
2. Solve for the radius squared (r²).
3. Write the equation with the center and radius squared.
- Example: Write the equation of a circle with a center of (0, 0) and a point at (5, -12).
- (5-0)² + (-12-0)² = r² → 25 + 144 = r² → r² = 169.
- Equation: x² + y² = 169.
- Example: Write the equation of a circle with a center of (2, -3) and a point at (-4, 5).
- (-4-2)² + (5-(-3))² = r² → (-6)² + (8)² = r² → 36 + 64 = r² → r² = 100.
- Equation: (x-2)² + (y+3)² = 100.
- Try: Write the equation of a circle with a center of (-5, -7) and a point at (3, 0).
- (3-(-5))² + (0-(-7))² = r² → (8)² + (7)² = r² → 64 + 49 = r² → r² = 113.
- Equation: (x+5)² + (y+7)² = 113.