circlescircles 11.5 equations of circles objective: to write an equation of a circle
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Objective:Objective: To write an equation of a circle.
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Equation of a Circle
The center of a circle is given by (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
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Ex. 1: Writing a Standard Equation of a Circle
• Write the standard equation of the circle with a center 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 Substitute values.
(x + 4)2 + (y – 0)2 = 50.41 Simplify.
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Example 2:Writing Equations of Circles
Write the standard equation of the circle:
Center (-3, 8) Radius of 6.2
(x + 3)2 + (y – 8)2 = 38.44
(x – h)2 + (y – k)2 = r2
[(x – (-3)]2 + (y – 8)2 = 6.22
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Example 3: Writing Equations of Circles
Write the standard equation of the circle:
Center (0, 6) Radius of
x 2 + (y – 6)2 = 7
7
(x – h)2 + (y – k)2 = r2
[(x – 0]2 + (y – 6)2 = (√7)2
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Example 4: Writing Equations of Circles
Write the standard equation of the circle:
Center (-1.9, 8.7) Radius of 3
(x + 1.9)2 + (y – 8.7)2 = 9
(x – h)2 + (y – k)2 = r2
[(x – (-1.9)]2 + (y – 8.7)2 = 32
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Ex. 5: Writing a Standard Equation of a Circle• The point (1, 2) is on a circle whose center is (5,
-1). Write a standard equation of the circle.
212
212 )()( yyxx r =
r = 22 )21()15(
r = 22 )3()4( r = 916 r = 25
r = 5
Use the Distance Formula
Substitute values.
Simplify.
Simplify.
Addition Property
Square root the result.
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Ex. 5 (cont.): Writing a Standard Equation of a Circle• The point (1, 2) is on a circle whose center is (5,
-1). Write a standard equation of the circle.
(x – h)2 + (y – k)2 = r2 Standard equation of a circle.
[(x – 5)]2 + [y –(-1)]2 = 52 Substitute values.
(x - 5)2 + (y + 1)2 = 25 Simplify.
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Ex. 6: Writing a Standard Equation of a Circle• The point (-1,-4) is on a circle whose center is (-
4, 0). Write a standard equation of the circle.
212
212 )()( yyxx r =
r = 22 )04())4(1(
r = 22 )4()3( r = 169 r = 25
r = 5
Use the Distance Formula
Substitute values.
Simplify.
Simplify.
Addition Property
Square root the result.
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Ex. 6 (cont.): Writing a Standard Equation of a Circle• The point (-1, -4) is on a circle whose center is
(-4, 0). Write a standard equation of the circle.
(x – h)2 + (y – k)2 = r2 Standard equation of a circle.
[(x – (-1))]2 + [y –(-4)]2 = 52 Substitute values.
(x - 1)2 + (y + 4)2 = 25 Simplify.
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Example 7: Give the center and radius of the following circles:
A.
B.
16)3()4( 22 yx
1)7( 22 yx
Center: ( 4, 3 ) ; Radius: 4
Center: ( 0, -7 ) ; Radius: 1
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Graphing Circles
• If you know the equation of a circle, you can graph the circle by identifying its center and radius.
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Ex. 8: Graphing a circle
• The equation of a circle is
(x+2)2 + (y-3)2 = 9. Graph the circle.
First rewrite the equation to find the center and its radius.
• (x+2)2 + (y-3)2 = 9• [x – (-2)]2 + (y – 3)2=32
• The center is (-2, 3) and the radius is 3.
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Ex. 8 (cont.): Graphing a circle
• To graph the circle, place a point at (-2, 3), set the radius at 3 units.
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Example 9: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
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Example 10: 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
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Example 11: 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
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Example 12: Graphing Circles
(x – 3)2 + (y – 2)2 = 9
Center (3, 2)
Radius of 3
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Example 13: Graphing Circles
(x + 4)2 + (y – 1)2 = 25
Center (-4, 1)
Radius of 5
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