1. GeometryGeometry 10.6 Equations of Circles Geometry

2. GeometryGeometry Objectives Write the equation of a circle. Use the equation of a circle and its graph to solve problems.

3. GeometryGeometry Finding Equations of Circles You can write an equation of a circle in a coordinate plane if you know its radius and the coordinates of its center.

4. GeometryGeometry Finding Equations of Circles Suppose the radius is r and the center is (h, k). Let (x, y) be any point on the circle. The distance between (x, y) and (h, k) is r, so We use Distance Formula. (Told you it wasn’t going away).

5. GeometryGeometry Finding Equations of Circles Square both sides to find the standard equation of a circle with radius r and center (h, k). (x – h) 2 + (y – k) 2 = r 2 If the center is at the origin, then the standard equation is x 2 + y 2 = r 2.

6. GeometryGeometry Writing a Standard Equation of a Circle Write the standard equation of the circle with a center at (-4, 0) and radius 7 (x – h) 2 + (y – k) 2 = r 2 Standard equation of a circle. [(x – (-4)] 2 + (y – 0) 2 = 7 2 Substitute values. (x + 4) 2 + (y – 0) 2 = 49 Simplify.

7. GeometryGeometry 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.

8. GeometryGeometry Ex. 2: 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.

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

10. GeometryGeometry 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.

11. GeometryGeometry To graph the circle, place the point of a compass at (-2, 3), set the radius at 3 units, and swing the compass to draw a full circle.

12. GeometryGeometry Graphs of Circles 1.Rewrite the equation to find the center and radius. The center is at (13, 4) and the radius is 4. –(x – h) 2 + (y – k) 2 = r 2 –(x – 13) 2 + (y – 4) 2 = 4 2 –(x - 13) 2 + (y - 4) 2 = 16

13. GeometryGeometry The center is (1, -3) Radius = 3

14. GeometryGeometry The center is (-4, -3) Radius = 1

15. GeometryGeometry (x – h) 2 + (y – k) 2 = r 2 (x – 8) 2 + (y + 6) 2 = 4 2 (x – 8) 2 + (y +6) 2 =16 Find Radius 3.14 = 3.14 * r 2 r = 1 (x – h) 2 + (y – k) 2 = r 2 (x + 12) 2 + (y +11) 2 = 1 2 (x +12) 2 + (y +11) 2 =1

16. GeometryGeometry Find Radius 12 * 3.14 = 2 *3.14 * r r = 16 (x – h) 2 + (y – k) 2 = r 2 (x + 9) 2 + (y +9) 2 = 6 2 (x +12) 2 + (y +11) 2 =36 Find Radius 8 * 3.14 = 2 *3.14 * r r = 4 (x – h) 2 + (y – k) 2 = r 2 (x + 13) 2 + (y -4) 2 = 4 2 (x +13) 2 + (y - 4) 2 =16

17. GeometryGeometry (x – h) 2 + (y – k) 2 = r 2 (x - 4) 2 + (y + 0) 2 = 4 2 (x – 4) 2 + y 2 =16 The center is (4, 0) Radius = 3

18. GeometryGeometry The center is (-3,-4) Radius = 2 (x – h) 2 + (y – k) 2 = r 2 (x +3) 2 + (y + 4) 2 = 2 2 (x + 3) 2 + (y+4) 2 =4

19. GeometryGeometry The center is (2,4) Radius = √6 = 2.5

20. GeometryGeometry The center is (4,3) Radius = √2 = 1.4