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10.6 Equations of Circles

Geometry

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Objectives

• Write the equation of a circle.• Use the equation of a circle and its

graph to solve problems.

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Finding Equations of Circles

• You can write an equation of a circle in a coordinate plane if you know its radius and the coordinates of its center.

6

4

2

-2

5 10 15x

y

(h, k)

(x, y)

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Finding Equations of Circles• Suppose the radius

is r and the center is (h, k).

• Let (x, y) be any point on the circle.

• The distance between (x, y) and (h, k) is r, so

• We use Distance Formula.

• (Told you it wasn’t going away).

6

4

2

-2

5 10 15x

y

(h, k)

(x, y)

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Finding Equations of Circles

• Square both sides to find the standard equation of a circle with radius r and center (h, k).

(x – h)2 + (y – k)2 = r2

If the center is at the origin, then the standard equation is

x2 + y2 = r2.

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5 10 15x

y

(h, k)

(x, y)

rkyhx 22 )()(

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Writing a Standard Equation of a Circle

• Write the standard equation of the circle with a center at (-4, 0) and radius 7

(x – h)2 + (y – k)2 = r2 Standard equation of a circle.

[(x – (-4)]2 + (y – 0)2 = 72 Substitute values.

(x + 4)2 + (y – 0)2 = 49 Simplify.

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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 = 916r = 25

r = 5

Use the Distance Formula

Substitute values.

Simplify.

Simplify.

Addition Property

Square root the result.

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Ex. 2: Writing a Standard Equation of a CircleThe 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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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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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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Graphing a circle

• To graph the circle, place the point of a compass at (-2, 3), set the radius at 3 units, and swing the compass to draw a full circle.

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-5 5

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Graphs of Circles

1. Rewrite the equation to find the center and radius. The center is at (13, 4) and the radius is 4.

– (x – h)2 + (y – k)2= r2

– (x – 13)2 + (y – 4)2= 42

– (x - 13)2 + (y - 4)2 = 16

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• The center is (1, -3)• Radius = 3

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• The center is (-4, -3)• Radius = 1

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• (x – h)2 + (y – k)2 = r2

• (x – 8)2 + (y + 6)2 = 42

• (x – 8)2 + (y +6)2 =16

• Find Radius

3.14 = 3.14 * r2

• r = 1 • (x – h)2 + (y – k)2 = r2

• (x + 12)2 + (y +11)2 = 12

• (x +12)2 + (y +11)2 =1

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• Find Radius

12 * 3.14 = 2 *3.14 * r

• r = 16 • (x – h)2 + (y – k)2 = r2

• (x + 9)2 + (y +9)2 = 62

• (x +12)2 + (y +11)2 =36

• Find Radius

8 * 3.14 = 2 *3.14 * r

• r = 4 • (x – h)2 + (y – k)2 = r2

• (x + 13)2 + (y -4)2 = 42

• (x +13)2 + (y - 4)2 =16

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• (x – h)2 + (y – k)2 = r2

• (x - 4)2 + (y + 0)2 = 42

• (x – 4)2 + y 2 =16

• The center is (4, 0)• Radius = 3

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• The center is (-3,-4)• Radius = 2

• (x – h)2 + (y – k)2 = r2

• (x +3)2 + (y + 4)2 = 22

• (x + 3)2 + (y+4) 2 =4

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• The center is (2,4)• Radius = √6 = 2.5

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• The center is (4,3)• Radius = √2 = 1.4