Geometry 10.6 Equations of a Circle [email protected].

Geometry

10.6 Equations of a Circle

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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.



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.



Circle Equation

r(x, y)

(h, k)

Use the Distance Formula to write this.

2 2( ) ( )r x h y k



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



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



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




(x – 2)2 + (y + 1)2 = 1 (x – 2)2 + (y – (-1))2 = 12

Center: (2, -1) Radius: 1




(x – 6)2 + y2 = 100

Center: (6, 0) Radius: 10 (x – 6)2 + (y – 0)2 = 102



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



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



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



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

(5, 4). Write the equation.



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


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!



Write the equation. The point (3, 2) is on a circle with center


2 2 2( ) ( )r x h y k

r2 = 8

(x – 5)2 + (y – 4)2 = 8



Your Turn. The point (-1, 4) is on a circle with center


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



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



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



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.



Graphing Circles


1)Draw the center.


3)Sketch the circle.



Your Turn

Graph:

(x – 1)2 + (y + 3)2 = 16

Solution:

Center: (1, -3)

Radius: 4



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.



Graphical Solution

Graph:

(x + 1)2 + (y – 1)2 = 25

Solution:

Center: (-1, 1)

Radius: 5

Locate (3, 4)

On the circle.



What about (3, 2)?

In the interior of the circle.



What about (-5, -3)?

In the exterior of the circle.



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



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.



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



Quick Practice

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

Center (-2, 0) Radius = 3



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



Quick Practice

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



Homework

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Geometry 10.6 Equations of a Circle [email protected].

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