1. Geometry
Equations of Circles

2. What You Should Learn Why You Should Learn It
How to find the equation of a circle
How to use the equation of a circle to solve problems
You can use equations of circles to solve real-life problems such as drawing plans for a rotary engine

3. Finding Equations of Circles
Remember the distance formula?
Consider a circle in a coordinate plane
Center is (h,k)
(x,y) is any point on the circle
Distance between (h,k) and (x,y) is r
Using the distance formula

4. Standard Equation of a Circle
The standard equation of a circle with radius r and center (h,k) is (x – h)2 + (y – k)2 = r2
If the center of the circle is the origin, then the standard equation has the simpler form x2 + y2 = r2

5. Example 1 Writing a Standard Equation of a Circle
Write the standard equation of the circle whose center is (3,-1) and whose radius is 4.

6. Example 1 Writing a Standard Equation of a Circle
Write the standard equation of the circle whose center is (3,-1) and whose radius is 4.
(h,k) = (3,-1) and r = 4
(x – h)2 + (y – k)2 = r2
(x – 3)2 + (y – -1)2 = 42
(x – 3)2 + (y + 1)2 = 16

7. Example 2 Writing a Standard Equation of a Circle
The point (1,2) is on a circle whose center is (0,0), Write its standard equation.
Hint: first find the radius by using the distance formula

8. Example 2 Writing a Standard Equation of a Circle
The point (1,2) is on a circle whose center is (0,0), Write its standard equation.
(x1,y1) = (0,0) and (x2,y2) = (1,2)
x2 + y2 = r2

9. Any 3 noncollinear points determine a circle
Procedure for finding the equation of a circle that passes through three points
1. Consider the Δ formed by the 3 points. Draw the perpendicular bisector of two sides
2. The center of the circle is the point of intersection of the two perpendicular bisectors
3. The radius of the circle is the distance between the center and any of the three given points
4. Use the center and radius to write the standard equation of the circle

10. Example 3 A Circle Passing through Three Points
Find an equation of the circle that passes through A(-1,5), B(7,1), and C(5,-3) 4 2 -2 2 8 -2

11. Example 3 Solution Step 1
Find an equation of the circle that passes through A(-1,5), B(7,1), and C(5,-3)
Draw the perpendicular bisector of AB
Thus, the perpendicular bisector of AB is L1, whose slope is 2 and passes through the point (3,3)

12. Example 3 Solution Step 1
Find an equation of the circle that passes through A(-1,5), B(7,1), and C(5,-3)
Draw the perpendicular bisector of BC
Thus, the perpendicular bisector of BC is L2, whose slope is -½ and passes through the point (6,-1)

13. Example 3 Solution Step 2
Find an equation of the circle that passes through A(-1,5), B(7,1), and C(5,-3)
The center of the circle is the point where the two perpendicular bisectors meet.
Center of Circle = (2,1)

14. Example 3 Solution Step 3
Find an equation of the circle that passes through A(-1,5), B(7,1), and C(5,-3)
The radius of the circle is the distance between (2,1) and any of the original points
(x1,y1) = (2,1) and (x2,y2) = (7,1)

15. Example 3 Solution Step 4 (x – h)2 + (y – k)2 = r2
Find an equation of the circle that passes through A(-1,5), B(7,1), and C(5,-3)
Radius = 5 and center (h,k) = (2,1)
(x – h)2 + (y – k)2 = r2
(x – 2)2 + (y – 1)2 = 52
(x – 2)2 + (y – 1)2 = 25

16. Example 4

17. Example 4 Solution

18. THE END