2. Section 9-3 Circles

3. Objectives I can write equations of circles I can graph circles with certain properties I can Complete the Square to get into Standard Format

4. Circles A circle is a set of points in a plane equal distance from a center point. The distance from the center point to any part of the circle is called the radius. r

5. Circle Equation The equation for any circle with center (h, k) and radius r units is given by the following: (x – h) 2 + (y – k) 2 = r 2

6. Example 1 Write an equation of a circle with center point at (-6, 11) and radius of 6 units (x - h) 2 + (y – k) 2 = r 2 (Basic Formula) h = -6 k = 11 r = 6 ( x - -6) 2 + (y – 11) 2 = 6 2 (x + 6) 2 + (y – 11) 2 = 36

7. Example 2 Write an equation of a circle with center point at (3, -4) and diameter of 6 units (x - h) 2 + (y – k) 2 = r 2 (Basic Formula) h = 3 k = -4 If diameter is 6 units then r = 3 ( x - 3) 2 + (y - -4) 2 = 3 2 (x - 3) 2 + (y + 4) 2 = 9

8. Example 3 Write an equation of a circle with center point at (1, 4) and radius of units (x - h) 2 + (y – k) 2 = r 2 (Basic Formula) h = 1 k = 4 r = ( x - 1) 2 + (y - 4) 2 = 2 (x - 1) 2 + (y - 4) 2 = 7

9. Example : Working Backwards Find the center and radius of the circle with an equation when you need to complete the square: x 2 + y 2 + 16x – 22y – 20 = 0 We want the equation (x - h) 2 + (y - k) 2 = r 2 (x 2 + 16x) + (y 2 – 22y) = 20 (x 2 + 16x + ___) + (y 2 –22y + ___) = 20 + __ + ___ (x 2 + 16x + 64) + (y 2 – 22y + 121) = 20 + 64 + 121 (x + 8)(x + 8) + (y – 11)(y – 11) = 205 (x + 8) 2 + (y – 11) 2 = 205 So Center is (-8, 11) Radius: r 2 = 205, so r = 14.3 units

10. Homework Worksheet 10-2