Geometry

Equations of a Circle

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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.

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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.

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Circle Equation

r(x, y)

(h, k)

Use the Distance Formula to write this.

2 2( ) ( )r x h y k

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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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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

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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

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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

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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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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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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

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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

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What if we don’t know r? The point (3, 2) is on a circle with center

(5, 4). Write the equation.

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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!

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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

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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

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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

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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

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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.

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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.

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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.

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Your Turn

Graph:

(x – 1)2 + (y + 3)2 = 16

Solution:

Center: (1, -3)

Radius: 4

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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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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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What about (3, 2)?

In the interior of the circle.

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What about (-5, -3)?

In the exterior of the circle.

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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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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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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

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Quick Practice

1. Identify the center and the radius of the circle: (x + 2)2 + y2 = 9

Center (-2, 0) Radius = 3

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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

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Quick Practice

3. Sketch the graph of the circle given by the equation (x - 1)2 + (y + 3)2 = 1

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Practice Problems