Post on 27-Mar-2015
Geometry
10.6 Equations of a Circle
mbhaub@mpsaz.org
Thursday, March 26, 2:44
Geometry 10.6 Equations of Circles 2
Goals
Write the equation of a circle. Use the equation of a circle to graph
the circle on the coordinate plane. Solve problems with circles.
Thursday, March 26, 2:44
Geometry 10.6 Equations of Circles 3
Circle Definition
A circle is the set of points on a plane that are equidistant from the center.
r(x, y)
(h, k)
The radius, r, is the distance between the center (h, k) and any point (x, y) on the circle.
Thursday, March 26, 2:44
Geometry 10.6 Equations of Circles 4
Circle Equation
r(x, y)
(h, k)
Use the Distance Formula to write this.
2 2( ) ( )r x h y k
Thursday, March 26, 2:44
Geometry 10.6 Equations of Circles 5
Circle Equation
2 2 2( ) ( )r x h y k r
(x, y)
(h, k)
Square both sides:
2 2( ) ( )r x h y k
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Geometry 10.6 Equations of Circles 6
The Equation of a Circle
2 2 2( ) ( )r x h y k
r(x, y)
(h, k)
Where:
(h, k) is the center
r is the radius
(x, y) is any point on the circle
Thursday, March 26, 2:44
Geometry 10.6 Equations of Circles 7
What is the center and radius?
(x – 9)2 + (y – 1)2 = 25
Center: (9, 1) Radius: 5 (x – 9)2 + (y – 1)2 = 52
Thursday, March 26, 2:44
Geometry 10.6 Equations of Circles 8
What is the center and radius?
(x – 2)2 + (y + 1)2 = 1 (x – 2)2 + (y – (-1))2 = 12
Center: (2, -1) Radius: 1
Thursday, March 26, 2:44
Geometry 10.6 Equations of Circles 9
What is the center and radius?
(x – 6)2 + y2 = 100
Center: (6, 0) Radius: 10 (x – 6)2 + (y – 0)2 = 102
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Geometry 10.6 Equations of Circles 10
Your Turn
Identify the center and radius of each circle:
(x – 12)2 + (y + 3)2 = 4 Center: (12, –3) Radius = 2 x2 + y2 = 121 Center: (0, 0) Radius = 11
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Geometry 10.6 Equations of Circles 11
Example Write the equation of a circle with center
(5, 6) and radius 4.
2 2 2( ) ( )r x h y k
42 = (x – 5)2 + (y – 6)2
16 = (x – 5)2 + (y – 6)2
or (x – 5)2 + (y – 6)2 = 16
Thursday, March 26, 2:44
Geometry 10.6 Equations of Circles 12
Your Turn Write the equation of a circle with center
(1, -3) and radius 8.
2 2 2( ) ( )r x h y k
82 = (x – 1)2 + (y – (-3))2
(x – 1)2 + (y + 3)2 = 64
Thursday, March 26, 2:44
Geometry 10.6 Equations of Circles 13
What if we don’t know r? The point (3, 2) is on a circle with center
(5, 4). Write the equation.
Thursday, March 26, 2:44
Geometry 10.6 Equations of Circles 14
What if we don’t know r? The point (3, 2) is on a circle with center
(5, 4). Write the equation.
2 2 2( ) ( )r x h y k
r2 = (3 – 5)2 + (2 – 4)2
r2 = (–2 )2 + (–2)2
r2 = 4 + 4 = 8DON’T SIMPLIFY!
Thursday, March 26, 2:44
Geometry 10.6 Equations of Circles 15
Write the equation. The point (3, 2) is on a circle with center
(5, 4). Write the equation.
2 2 2( ) ( )r x h y k
r2 = 8
(x – 5)2 + (y – 4)2 = 8
Thursday, March 26, 2:44
Geometry 10.6 Equations of Circles 16
Your Turn. The point (-1, 4) is on a circle with center
(2, 3). Write the equation.
2 2 2( ) ( )r x h y k r2 = (-1 – 2)2 + (4 – 3)2
r2 = (-3)2 + (1)2
r2 = 9 + 1 = 10
(x – 2)2 + (y – 3)2 = 10
Thursday, March 26, 2:44
Geometry 10.6 Equations of Circles 17
Graphing Circles
Graph the circle given by the equation
(x – 2)2 + (y – 1)2 = 9
First find the center (h, k).
What is h?
2
What is k?
1
Thursday, March 26, 2:44
Geometry 10.6 Equations of Circles 18
Graphing Circles continued
(x – 2)2 + (y – 1)2 = 9
Center (2, 1)
What is r?
3
Why?
(x – 2)2 + (y – 1)2 = 32
Thursday, March 26, 2:44
Geometry 10.6 Equations of Circles 19
Graphing Circles
Knowing the center is (2, 1) and the radius is 3. Graph the circle.
1)Draw the center.
2)Draw points at the ends of 4 radii.
Thursday, March 26, 2:44
Geometry 10.6 Equations of Circles 20
Graphing Circles
Knowing the center is (2, 1) and the radius is 3. Graph the circle.
1)Draw the center.
2)Draw points at the ends of 4 radii.
3)Sketch the circle.
Thursday, March 26, 2:44
Geometry 10.6 Equations of Circles 21
Graphing Circles
Knowing the center is (2, 1) and the radius is 3. Graph the circle.
1)Draw the center.
2)Draw points at the ends of 4 radii.
3)Sketch the circle.
Thursday, March 26, 2:44
Geometry 10.6 Equations of Circles 22
Your Turn
Graph:
(x – 1)2 + (y + 3)2 = 16
Solution:
Center: (1, -3)
Radius: 4
Thursday, March 26, 2:44
Geometry 10.6 Equations of Circles 23
Problem
(x + 1)2 + (y – 1)2 = 25Is the point (3, 4) on the circle, in its interior,
or in the exterior?
Directions: Make a sketch of the circle. Then locate (3, 4) and answer the question.
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Geometry 10.6 Equations of Circles 24
Graphical Solution
Graph:
(x + 1)2 + (y – 1)2 = 25
Solution:
Center: (-1, 1)
Radius: 5
Locate (3, 4)
On the circle.
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Geometry 10.6 Equations of Circles 25
What about (3, 2)?
In the interior of the circle.
Thursday, March 26, 2:44
Geometry 10.6 Equations of Circles 26
What about (-5, -3)?
In the exterior of the circle.
Thursday, March 26, 2:44
Geometry 10.6 Equations of Circles 27
You could do this…
Since the distance to the point is larger than the radius, it must be in the exterior of the circle.
Find the distance from the center (-1, 1) to the point (-5, -3):
2 2
2 2
( 1 ( 5)) (1 ( 3))
4 4 32
5.65
d
5
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Geometry 10.6 Equations of Circles 28
What you can now do:
Write the equation of a circle. Graph a circle from its equation. Determine where a point is in the
interior, exterior, or on a circle.
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Geometry 10.6 Equations of Circles 29
Quick Practice
1. Identify the center and the radius of the circle: (x + 2)2 + y2 = 9
2. Find the equation of a circle if the center is (1, 2) and the point (3, 0) is on the circle.
3. Sketch the graph of the circle given by the equation (x - 1)2 + (y + 3)2 = 1
Thursday, March 26, 2:44
Geometry 10.6 Equations of Circles 30
Quick Practice
1. Identify the center and the radius of the circle: (x + 2)2 + y2 = 9
Center (-2, 0) Radius = 3
Thursday, March 26, 2:44
Geometry 10.6 Equations of Circles 31
Quick Practice
2. Find the equation of a circle if the center is (1, 2) and the point (3, 0) is on the circle.
2
2 2
2 2
2
2
( 1) ( 2
(3 1) (
)
0 )
4 4
8
8
2 r
r
x y
r
Thursday, March 26, 2:44
Geometry 10.6 Equations of Circles 32
Quick Practice
3. Sketch the graph of the circle given by the equation (x - 1)2 + (y + 3)2 = 1
Thursday, March 26, 2:44
Geometry 10.6 Equations of Circles 33
Homework
mbhaub@mpsaz.org