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6-1 Circles

Unit 6 Conics

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Concepts and Objectives

Circles (Obj. #19)

Identify the equation of a circle. Write the equation of a circle, given the center and

the radius.

se t e comp et ng t e square met o to eterm nethe center and radius of a circle.

Write the equation of a circle, given the center and a

point on the circle.

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Graphing Circles

To graph a circle on graph paper, plot the center point

and count out the radius. Open the compass to thatpoint and draw the circle.

2 2

Center: (3, 4),

radius =

=

=9 3

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Graphing Circles

Enter

Enter

Change theq setting to square

( )= +2

9 3 4y x

( )= +29 3 4y x

The calculator may show a gap, butthats okay.

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General Form of a Circle

Suppose we took the center-radius equation and

expanded the binomials , and set everything equal to 0:

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

+ + + =2 2 2 2 22 2 0 x hx h k k r

If we letc = 2h, d= 2k, and e = h2 + k2 r2, we have

( ) ( ) ( )+ + + + + =2 2 2 2 22 2 0x y h x k y h k r

+ + + + =2 2

0 x y cx dy e

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General Form of a Circle

To get from the general equation back to the center-

radius form (so we can know the center and the radius),we complete the square for bothxandy.

whose equation is

+ + =2 2 4 8 44 0 x y x y

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General Form of a Circle

Example: What is the center and radius of the circle

whose equation is+ + =2 2 4 8 44 0 x y x y

( ) ( )+ + =2 24 8 44 x x y y

The center is at(2, 4), and the radius is 8.

+ + =

+ + + +

22

22

4 8 444 8

4 162 2

x x y y

( ) ( )+ + + + =2 2 2 24 2 8 4 64 x x y y

( ) ( )+ + =2 2 2

2 4 8x y

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General Form of a Circle

Example: What is the center and radius of the circle

whose equation is+ + =2 22 2 2 6 45 0 x y x y

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General Form of a Circle

Example: What is the center and radius of the circle

whose equation is+ + =2 22 2 2 6 45 0 x y x y

( ) ( ) + + =2 22 2 3 45 x x y y

In order to complete the

square, the coefficients of

the square term must be 1.

Dont forget

to distribute!

+ + = + + + +

2 2

2 2

2 2 3 451 3 1 9

2 2 2 2 x x y y

+ + =

2 2

1 3

2 2 502 2x y

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General Form of a Circle

Example (cont.):

+ + =

2 21 3

2 2 502 2

x yDivide through

by 2.

The center is at , and the radius is 5.

+ + =

2 2

1 3 252 2

x y

1 3

,2 2

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Characteristics Example: The graph of the equation

is either a point or is nonexistent. Which is it?+ + + =

2 2

8 2 24 0 x y x y

2 2

2 28 2

Since r2 is negative, the graph is nonexistent.

2 2

( ) ( ) + + + + = 2 2 2 28 4 2 1 7 x x y y

( ) ( ) + + = 2 2

4 1 7x y

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Writing the Equation of a CircleWe can now tell that the equation

is a circle with a center at(2, 3) and a radius of 4.

Suppose I wanted to know whether (6, 3) was on the

( ) ( ) + =2 2

2 3 16x y

c rc e. ow cou n out In order to be on the circle, the point must satisfy the

equation. That is, if we plug in 6 forxand 3 fory, and

we get 16, the point is on the circle.

( ) ( ) + =2 2

6 2 3 3 16?

=16 16

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Writing the Equation of a Circle We can use this idea to write the equation of a circle

given the center and a point on the circle.

Example: Write the equation of the circle with center at

, , .

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Writing the Equation of a Circle Example: Write the equation of the circle with center at

(4, 5) that contains the point(2, 3).

( ) ( )+ + =2 2 2

4 5 x y r

=2 2 2

Therefore, the equation of the circle is

= 28 r

( ) ( )+ + =

2 2

4 5 8x y Dont square the8its already

squared!

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Circular Inequalities Circular inequalities are fairly straightforward. For the

center-radius form of the circle, the graph will be The region inside the circle if the symbol is or

s w t nes, < or > s grap e w t a otte ne an

or is graphed with a solid line.

<

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Homework College Algebra (brown book)

Page 199: 21-30 (3s) Turn in: 21, 24

Algebra & Trigonometry(green book)

Page 466: 2-18 (even) Turn in: 4, 6, 12, 14