1. Circles: Objectives/Assignment
Write the equation of a circle.
Use the equation of a circle and its graph to solve problems.
Graphing a circle using its four quick points.

2. What do you know about circles?

3. Definitions
Circle: The set of all points that are the same distance (equidistant) from a fixed point.
Center: the fixed points
Radius: a segment whose endpoints are the center and a point on the circle

4. The equation of circle centered at (0,0) and with radius r
Solution:
Let P(x, y) represent any point on the circle
x 2 + y 2 = r 2

5. Finding the Equation of a Circle
The center is (0, 0) The radius is
The equation is:
x 2 + y 2 = 144

6. Write out the equation for a circle centered at (0, 0) with radius =1
Solution:
Let P(x, y) represent any point on the circle

7. Equation of a Circle

8. Ex. 1: Writing a Standard Equation of a Circle centered at (-4, 0) and radius 7.1
(x – h)2 + (y – k)2 = r2 Standard equation of a circle.
[(x – (-4)]2 + (y – 0)2 =7.12 Substitute values.
(x + 4)2 + y2 = Simplify.

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

10. The equation of the circle is:
(x –5)2 +[y –(-1)]2 = 52
(x –5)2 +(y + 1)2 = Simplify

11. You try!!: Write the equation of the circle :
(x-5)2 + ( y+4)2 = 49
(x+3)2 + ( y- 6)2 = 25

12. THINK ABOUT IT
Find the center, the length of the radius, and write the equation of the circle if the endpoints of a diameter are (-8,2) and (2,0).
Center: Use midpoint formula!
Length: use distance formula with radius and an endpoint
Equation: Put it all together

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

14. 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

15. Finding the Equation of a Circle
Circle O
The center is (0, 0)
The radius is
The equation is
x 2 + y 2 = 144

16. Graphing Circles
If you know the equation of a circle, you can graph the circle
by identifying its center and radius;
By listing four quick points: the upmost, lowest, leftmost and rightmost points.

17. Graphing Circles Using 4 quick points
(x – 3)2 + (y – 2)2 = 9
Center (3, 2)
Radius of 3
Leftmost point (0, 2)
Rightmost point(6, 2)
Highest point(3, 5)
Lowest point(3, -1)

18. Graphing Circles by listing its four quick points
(x + 4)2 + (y – 1)2 = 25
Center (-4, 1)
Radius of 5
Left: (-9, 1)
Right: (1, 1)
Up: (-4, 6)
Low: (-4, -4)

19. Ex. 4: Applying Graphs of Circles
A bank of lights is arranged over a stage. Eachlight illuminates a circular area on the stage. A coordinate plane is used to arrange the lights, using the corner of the stage as the origin. The equation (x – 13)2 + (y - 4)2 = 16 represents one of the disks of light. A. Graph the disk of light. B. Three actors are located as follows: Henry is at (11, 4), Jolene is at (8, 5), and Martin is at (15, 5). Which actors are in the disk of light?

20. Ex. 4: Applying Graphs of Circles
1. Rewrite the equation to find the center and
radius.
– (x – h)2 + (y – k)2= r2
– (x - 13)2 + (y - 4)2 = 16
– (x – 13)2 + (y – 4)2= 42
– The center is at (13, 4) and the radius is 4. The
circle is shown on the next slide.

21. Ex. 4: Applying Graphs of Circles
Graph the disk of light
The graph shows that Henry and Martin are both in the disk of light.

22. Ex. 4: Applying Graphs of Circles
A bank of lights is arranged over a stage. Each light is