Overview
This lecture covers how to identify and derive the equation of a circle, find its center and radius, and use completing the square to rewrite circle equations.
Equation of a Circle (Center at the Origin)
- The equation for a circle centered at the origin is ( x^2 + y^2 = r^2 ), where r is the radius.
- This formula is derived using Pythagoras' theorem for any point (x, y) on the circle.
Equation of a Circle (Center Not at the Origin)
- For a center at (a, b), the equation is ( (x - a)^2 + (y - b)^2 = r^2 ).
- Here, (x, y) is any point on the circle, and r is the radius.
- The terms ( x - a ) and ( y - b ) represent the horizontal and vertical distances from the center.
Completing the Square to Find Center and Radius
- Rearranging terms groups x and y parts together, e.g., ( x^2 + 8x + y^2 - 10y - 9 = 0 ).
- Complete the square separately for x and y terms.
- After completing the square, move constants to the right side to find the standard form.
- The center is found by making the squared brackets zero; the radius is the square root of the right side value.
Example Problems
- For ( x^2 + 8x + y^2 - 10y - 9 = 0 ), completing the square gives center (-4, 5), radius ( 5\sqrt{2} ).
- For ( x^2 + y^2 - 4x + 6y - 3 = 0 ), completing the square gives center (2, -3), radius 4.
Key Terms & Definitions
- Circle Equation (Origin) — ( x^2 + y^2 = r^2 ): circle center at (0, 0)
- Circle Equation (General) — ( (x - a)^2 + (y - b)^2 = r^2 ): circle center at (a, b)
- Completing the Square — a method to rewrite quadratics in standard form to identify circle properties
Action Items / Next Steps
- Complete questions from Exercise 5.2, focusing on both basic and advanced problems to build fluency with circle equations.