2. CirclesCircles 11.5 Equations of Circles Objective: To write an equation of a circle.

3. CirclesCircles Equation of a Circle The center of a circle is given by (h, k) The radius of a circle is given by r The equation of a circle in standard form is (x – h) 2 + (y – k) 2 = r 2

4. CirclesCircles Ex. 1: Writing a Standard Equation of a Circle Write the standard equation of the circle with a center at (-4, 0) and radius 7.1 (x – h) 2 + (y – k) 2 = r 2 Standard equation of a circle. [(x – (-4)] 2 + (y – 0) 2 = 7.1 2 Substitute values. (x + 4) 2 + (y – 0) 2 = 50.41 Simplify.

5. CirclesCircles Example 2:Writing Equations of Circles Write the standard equation of the circle: Center (-3, 8) Radius of 6.2 (x + 3) 2 + (y – 8) 2 = 38.44 (x – h) 2 + (y – k) 2 = r 2 [(x – (-3)] 2 + (y – 8) 2 = 6.2 2

6. CirclesCircles Example 3: Writing Equations of Circles Write the standard equation of the circle: Center (0, 6) Radius of x 2 + (y – 6) 2 = 7 (x – h) 2 + (y – k) 2 = r 2 [(x – 0] 2 + (y – 6) 2 = ( √7) 2

7. CirclesCircles Example 4: Writing Equations of Circles Write the standard equation of the circle: Center (-1.9, 8.7) Radius of 3 (x + 1.9) 2 + (y – 8.7) 2 = 9 (x – h) 2 + (y – k) 2 = r 2 [(x – (-1.9)] 2 + (y – 8.7) 2 = 3 2

8. CirclesCircles Ex. 5: Writing a Standard Equation of a Circle The point (1, 2) is on a circle whose center is (5, -1). Write a standard equation of the circle. r = r = 5 Use the Distance Formula Substitute values. Simplify. Addition Property Square root the result.

9. CirclesCircles Ex. 5 (cont.): Writing a Standard Equation of a Circle The point (1, 2) is on a circle whose center is (5, -1). Write a standard equation of the circle. (x – h) 2 + (y – k) 2 = r 2 Standard equation of a circle. [(x – 5)] 2 + [y –(-1)] 2 = 5 2 Substitute values. (x - 5) 2 + (y + 1) 2 = 25 Simplify.

10. CirclesCircles Ex. 6: Writing a Standard Equation of a Circle The point (-1,-4) is on a circle whose center is (- 4, 0). Write a standard equation of the circle. r = r = 5 Use the Distance Formula Substitute values. Simplify. Addition Property Square root the result.

11. CirclesCircles Ex. 6 (cont.): Writing a Standard Equation of a Circle The point (-1, -4) is on a circle whose center is (-4, 0). Write a standard equation of the circle. (x – h) 2 + (y – k) 2 = r 2 Standard equation of a circle. [(x – (-1))] 2 + [y –(-4)] 2 = 5 2 Substitute values. (x - 1) 2 + (y + 4) 2 = 25 Simplify.

12. CirclesCircles Example 7: Give the center and radius of the following circles: A. B. Center: ( 4, 3 ) ; Radius: 4 Center: ( 0, -7 ) ; Radius: 1

13. CirclesCircles Graphing Circles If you know the equation of a circle, you can graph the circle by identifying its center and radius.

14. CirclesCircles Ex. 8: Graphing a circle The equation of a circle is (x+2) 2 + (y-3) 2 = 9. Graph the circle. First rewrite the equation to find the center and its radius. (x+2) 2 + (y-3) 2 = 9 [x – (-2)] 2 + (y – 3) 2 =3 2 The center is (-2, 3) and the radius is 3.

15. CirclesCircles Ex. 8 (cont.): Graphing a circle To graph the circle, place a point at (-2, 3), set the radius at 3 units.

16. CirclesCircles Example 9:Finding the Equation of a Circle Circle A The center is (16, 10) The radius is 10 The equation is (x – 16) 2 + (y – 10) 2 = 100

17. CirclesCircles Example 10: Finding the Equation of a Circle Circle B The center is (4, 20) The radius is 10 The equation is (x – 4) 2 + (y – 20) 2 = 100

18. CirclesCircles Example 11: Finding the Equation of a Circle Circle O The center is (0, 0) The radius is 12 The equation is x 2 + y 2 = 144

19. CirclesCircles Example 12: Graphing Circles (x – 3) 2 + (y – 2) 2 = 9 Center (3, 2) Radius of 3

20. CirclesCircles Example 13: Graphing Circles (x + 4) 2 + (y – 1) 2 = 25 Center (-4, 1) Radius of 5

21. CirclesCircles Assignment Page 617 #1 - 23