LESSON 10-4
Equations of Circles
Created by Lisa Palen and Kristina Green
Henrico High School
Part I
Equations of Circles
Recall: Definitions
• Circle: The set of all points that are the
same distance from the center
• Radius: a segment whose endpoints are
the center and a point on the circle
• Radius: the LENGTH of a radius
Center
Radi us
Equation of a Circle
x y r
2
2
2
 x  h   y  k 
2
Center (0, 0)
Radius = r
2
r
2
Center (h, k)
Radius = r
Finding the Center and the Radius
when given the equation
1. x  y  25
2
2
Center (0, 0), r = 5
Center (0, 0), r = 10
2. x 2  y 2  100
3.  x  5    y  4   49 Center (5, -4), r = 7
2
2
4.  x  7    y  3  3 Center (-7, 3), r = 3
2
2
5. x   y  1  12
Center (0, 1), r = 12
6.  x  3  y 2  81
Center (3, 0), r = 9
2
2
2
Writing the Equation of a Circle
1.
2.
3.
4.
5.
6.
Center (0, 0) r = 2 x2 + y2 = 4
Center (0, 1) r = 6 x2 + (y – 1)2 = 36
Center (-3, 5) r = 2.5 (x + 3)2 + (y– 5)2= 6.25
2 + (y–10)2= 100
(x
+
5)
Center (-5, 10) r = 10
2 + y 2= 1
(x
–
8)
Center (8, 0) r = 1
2 + (y– 9)2= 11.56
(x–
6)
Center (6, 9) r = 3.4
Writing the Equation of a circle
2. A circle whose center is at (-3, 2) passes
through (-7, 2).
a. What is the length of the radius of the
circle?
b. Write the equation of the circle.
Answers:
a. r = 4
b. (x + 3)2 + (y - 2)2 = 16
Graphing
a
Circle
Find the
x  y 9
2
Answers:
center (0, 0)
radius = 3
2
center and the
radius and
graph the
circle.
Graphing
a
Circle
Find the
 x  1   y  2 
2
Answers:
center (1, -2)
radius = 5
2
 25
center and the
radius and
graph the
circle.
Graphing
a
Circle
Find the
2
x

3

y
4
 
2
Answers:
center (3, 0)
radius = 2
center and the
radius and
graph the
circle.
Writing the Equation of a circle
3. A circle has a diameter with endpoints
A (1, 2) and B (3, 6).
a. What is the center of the circle?
The midpoint of segment AB!
b. What is the radius of the circle?
The distance from the
center to A or B!
c. What is the equation of the circle?
Answers:
a. (2, 4)
b. sqrt (5)
c. (x – 2)2 + (y – 4)2 = 5
Finding the midpoint
For the last problem it was necessary to find
the midpoint, or the point halfway between two
points. There is a formula for this.
Part II
Midpoint
Reminder: What is a Midpoint?
• The midpoint of a segment AB is the point that divides
AB into two congruent segments.
• Where is the midpoint of AB?
A
Here
it is!
midpoint
B
Over
Here
?
Over
Here
?
Over
Here
?
Midpoint on a Number Line
• To find the midpoint of two points on a number line,
just average the coordinates.
ab
• Find the midpoint of GT.
2
G
T
x
• Take the average of the coordinates:
ab
2
4  9

2
5

2
= 2.5
midpoint
Finding a Midpoint in
The Coordinate Plane
We can find the midpoint between any two points in the
coordinate plane by finding the midpoint of the xcoordinates and the midpoint of the y-coordinates.
y
Example Find the
midpoint of the two
points.
midpoint?
x
Finding a Midpoint in
The Coordinate Plane
First: Find the average (midpoint) of the x-coordinates.
Remember: Take the average of the two coordinates.
a  b 4  8 4

 2
2
2
2
y
a  b 2  3 1

  0.5
2
2
2
–4
x
2
8
average of x-coordinates
Finding a Midpoint in
The Coordinate Plane
Next: Find the midpoint (average) of the y-coordinates.
Remember: Take the average of the two coordinates.
a  b 2  3 1

  0.5
2
2
2
y
a  b 2  3 1

  0.5
2
2
2
3
average of y-coordinates 0.5
x
– 22
average of x-coordinates
Finding a Midpoint in
The Coordinate Plane
Finally: The midpoint is the ordered pair:
(average of x-coordinates, average of y-coordinates)
y
= (2, 0.5)
a  b 2  3 1

  0.5
2
2
2
midpoint of y-coordinates 0.5
(2, 0.5)
x
2
midpoint of x-coordinates
The Midpoint Formula
The following formula combines what we did:
midpoint
= (average of x-coordinates, average of y-coordinates)
 x1  x2 y1  y2 

,

2 
 2
where (x1, y1) and (x2, y2) are the ordered pairs
corresponding to the two points.
So let’s go back to the example.
Example
Find the midpoint of the two points.
Solution: We already know the coordinates of the two points.
y
(8, 3)
midpoint?
x
(– 4, – 2)
Example
cont.
Solution cont.
Since the ordered pairs are
(x1, y1) = (-4, -2) and (x2, y2) = (8, 3)
Plug in x1 = -4, y1 = -2, x2 = 8 and y2 = 3 into
midpoint =
 x1  x2 y1  y2 
 2 , 2 


=
 4  8 2  3 
 2 , 2 


=
 4 1
 2,2


=
(2, 0.5)
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!
 8  2 2  0 
,

  3,1
2 
 2

Length: use distance formula
with center and an endpoint

(2  (3)) 2  (0  1) 2 
26
Equation: Put it all together
 x  (3)   ( y 1) 
2
2

26

2
or
 x  3
2
 ( y  1)  26
2