Transcript Equations of Circles

Circles:
Objectives/Assignment
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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.
CIRCLES
What do you know about circles?
Definitions
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Center
Radi us
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
The equation of circle centered at
(0,0) and with radius r
Solution:
Let P(x, y) represent any point on the circle
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1
x
2 +
y
2
=r
P
y
0.5
2
P
-2
x
-1
-0.5
P
-1
-1.5
1
P
2
Finding the Equation of a
Circle
The center is (0, 0) The radius is
The equation is:
x 2 + y 2 = 144
12
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
x  y 1
2
2
Equation of a Circle
C enter :  h , k 
R adius : r
x  h
2
yk  r
2
2
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
(x + 4)2 + y2 = 50.41
Substitute
Simplify.
values.
Ex. 2: Writing a Standard Equation of a
Circle The point (1, 2) is on a circle whose
center is (5,-1).
r 
x
r 
( 5  1)
2
 y
r 
4
r 
16  9
r 
25
r  5
2
2
 (1  2)
 (  3)
Use the Distance Formula
2
2
Substitute values.
2
Simplify
Addition Property
Square root the result.
The equation of the circle is:
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(x –5)2 +[y –(-1)]2 = 52
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(x –5)2 +(y + 1)2 = 25
Simplify
You try!!: Write the equation of the
circle :
Center : (5,  4)
Radius len gth: 7
(x-5)2 + ( y+4)2 = 49
Center : (  3, 6)
(x+3)2 + ( y- 6)2 = 25
Radius len gth: 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 
,


2 
 2
Length: use distance formula
with radius and an endpoint
  3,1 
(2  (  3))  (0  1) 
2
2
26
Equation: Put it all together
 x  (  3)   ( y  1) 
2
2

26

2
or
 x  3
2
 ( y  1)  26
2
Finding the Equation of a
Circle
Circle A
The center is (16, 10)
The radius is
10
The equation is
(x – 16)2 + (y – 10)2 = 100
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
Finding the Equation of a
Circle
Circle O
The center is (0, 0)
The radius is
12
The equation is
x 2 + y 2 = 144
Graphing Circles
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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.
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)
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)
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?
Ex. 4: Applying Graphs of
Circles
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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.
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.
Ex. 4: Applying Graphs of
Circles
A bank of lights is arranged over a stage. Each light is
r 
( 5  1)

4

16  9

25
 5
2
2
 (  3)
 (1  2)
2
2