Graph an Equation of a Circle

8/7/2019 Graph an Equation of a Circle

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WARM UP Conic Sections CA ST #16


2/19

Standard 16 Equation ofa circle with center at the origin.

The equation,in standard formofthe circle centered

inthe origin with radius ris:

x2 + y2 = r2 .

Example- x2 + y2 =25

Center is (0, 0)

Radius = 25 5!


3/19

Standard 16 Graphing the example

Center is (0, 0)

Radius = 25 5!

Example- x2 + y2 =25


4/19

Standard 16 Conic Sections CA ST #16

Equation ofa circle with center at (h, k).The equation,in standard formofthe circle with

center(h, k) andradius ris:

(x h)2

+ (y k)2 =

r2 .

Example- (x-3)2 + (y+2)2 = 9

Center is (3, -2)

Radius = 9 3!


5/19

Standard 16 Graphing the example

Center is (3, -2)

Radius = 9 3!

Example- (x-3)2 + (y+2)2 = 9

A (3, 1`)


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EXAMPLE1 Graph an equation ofa circle

Graph y2

= x2

+ 36. Identify the radius ofthe circle.

SOLUTION

STEP 1

Rewrite the equation y2

= x2

+ 36 instandard formas x2 + y2 = 36.

STEP 2

Identify the centerandradius. Fromthe equation,the

graph is a circle centeredatthe origin with radiusr = 36 = 6.


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EXAMPLE1 Graph an equation ofa circle

STEP 3Draw the circle. Firstplot several convenientpoints

thatare 6 units fromthe origin, such as (0, 6), (6, 0), (0,

6), and(6, 0). Thendraw the circle thatpasses

through the points.


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EXAMPLE2 Write an equation ofa circle

The point(2, 5) lies ona circle whose centeris theorigin. Write the standard formofthe equationofthe

circle.

SOLUTION

Because the point(2, 5) lies onthe circle,the circles

radius rmust be the distance betweenthe center(0, 0)

and(2, 5). Use the distance formula.

r = (2 0)2 + (5 0)2 = 29= 4 + 25

The radius is 29


9/19

EXAMPLE2 Write an equation ofa circle

Use the standard form with r to write anequationofthe circle.

= 29

x2 + y2 = r2 Standard form

x2 + y2 = ( 29 )2

Substitute forr29

x2 + y2 = 29 Simplify


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EXAMPLE3 Standardized Test Practice

SOLUTION

A line tangenttoa circle is

perpendiculartothe radius atthe

pointoftangency.Because the

radius tothe point (13, 2) has slope

= 2 0 3 0 =23

m


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EXAMPLE3 Standardized Test Practice

23

the slope ofthe tangent line at(3, 2) is thenegative reciprocal of or An equationof3

2the tangent line is as follows:

y 2 = (x (3))3

2

Point-slope form

32

y 2 = x + 92

Distributive property

32

132

y = x + Solve fory.

ANSWER

The correctansweris C.


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GUIDED PRACTICE for Examples 1, 2, and 3

Graph the equation. Identify the radius ofthe circle.

1. x2 + y2 = 9

SOLUTION 3


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2.

y2

= x2

+ 49SOLUTION 7


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3.x

2

18 = y2

SOLUTION 18


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4. Write the standard formofthe equationofthe circlethatpasses through (5, 1) and whose centeris the

origin.

SOLUTION x2 + y2 = 26

5. Write an equationofthe line tangenttothe circle

x2 + y2 = 37 at(6, 1).

y = 6x + 37SOLUTION


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EXAMPLE1 Graph the equation of circle with center (h. k)

Graph (x 2)

2

+ (y + 3)2

= 9.

SOLUTION

STEP 1

Compare the given equationtothe

standard formofan equationofa

circle. You can see thatthe graph is

a circle with centerat(h, k) = (2, 3)

andradius r = 9 = 3.


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EXAMPLE1 Graph the equation ofa translated circle

STE

P 2Plotthe center. Thenplot several points thatare each

3 units fromthe center:

(2 + 3, 3) = (5, 3) (2 3, 3) = (1, 3)

(2, 3 + 3) = (2, 0) (2, 3 3) = (2, 6)

STEP 3

Draw a circle through the points.


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GUIDED PRACTICE for Examples 1 and 2

Graph (x+ 1)

2

+ (y 3)2

= 4.

SOLUTION

circle with centerat(h, k) = ( 1, 3) andradius r = 2

1.


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Standard 16 Classwork/ Homework

Section 10-2 (page #435)Fromthe PH book

Problems 1-24 (yesoddsandevens)

Documents

Graph an Equation of a Circle