1. Objective: Write an equation of a Circle
Equations of Circles
Mr. Peter Richard
Wears Cool Glasses Too!
Objective: Write an equation of a Circle
And do other stuff too!

2. THE DISTANCE AND MIDPOINT FORMULAS
Investigating Distance: 1
Plot A(2,1) and B(6,4) on a coordinate plane. Then draw a right triangle that has AB as its hypotenuse.
B (6, 4)
A (2, 1) x y 2
Find and label the coordinates of the vertex C. 3
Find the lengths of the legs of
ABC. AB 5
yB – yA 3
4 – 1 4
Use the Pythagorean theorem to find AB.
Remember:
a 2 + b 2 = c 2
xB – xA
6 – 2 4
C (6, 1)
= c 2
= c
AB = 5
= c 2
25 = c

3. Finding the Distance Between Two Points
The steps used in the investigation can be used to develop a general formula for the distance between two points A(x 1, y 1) and B(x 2, y 2).
Using the Pythagorean theorem
x 2 – x 1
y2 – y1 d x y
C (x 2, y 1 )
B (x 2, y 2 )
A (x 1, y 1 )
a 2
+ b 2
= c 2
You can write the equation
(x 2 – x 1) 2 + ( y 2 – y 1) 2 = d 2
Solving this for d produces the
distance formula.
THE DISTANCE FORMULA
The distance d between the points (x 1, y 1) and (x 2, y 2) is
d = (x 2 – x 1) 2 + ( y 2 – y 1) 2

4. Compare with Distance Formula
Equation of a Circle
The center of a circle is (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 = r2
r = (x – h1) 2 + ( y – k) 2
Compare with Distance Formula
THE DISTANCE FORMULA
The distance d between the points (x 1, y 1) and (x 2, y 2) is
d = (x 2 – x 1) 2 + ( y 2 – y 1) 2

5. Given Center and Radius Write The Equation of the Circle
Center is (-6, 10) with radius of 4
(x – h)2 + (y – k)2 = r2
(x+6)2 + (y-10)2 = 16

6. Try some on your own Find the center of the circle given the equation
(9,-12)
(-9,12)

7. Example 2
Find the equation of a circle with center of (-3,7) and a radius of 8
(x-3)2+(y+7)2=64
(x-3)2+(y+7)2=8
(x+3)2+(y-7)2=64
(x+3)2+(y-7)2=8

8. (x – h)2 + (y – k)2 = r2 Writing an Equation of a Circle
Write the standard form of the equation of the circle that passes through the point (2,1) and whose center is the origin.
(h,k) = (0,0)
(x – h)2 + (y – k)2 = r2
This equation shows the pattern for all circles
Whose midpoint is at the origin.

9. Try This One Yourself!
Find the equation of a circle with a center of (-4,5) and has the point (3, 6)
(x-4)2+(y+5)2=50
(x-4)2+(y-5)2=49
(x+4)2+(y-5)2=48
(x+4)2+(y-5)2=50

10. r is the radius radius is 4 units Graphing an Equation of a Circle
Standard Equation of a Circle (Center at Origin)
r is the radius
radius is 4 units

11. Graphing an Equation of a Circle
Graph the equation. Give the radius of the circle.

12. Graphing an Equation of a Circle
Graph the equation. Give the radius of the circle.

13. Notes Over 10.3 Graphing an Equation of a Circle
Graph the equation. Give the radius of the circle.

14. Time to Circularize! Quiz: Page 564/565 # 20, 24, 28, 30, 50
Homework: # 19, 23, 27, 29, 49