CIrcles

8.3 Graph and Write Equations of Circles

Book Section 9.3

CirclesThe equation of a circle is:

222 )()( rkyhx

Where (h, k) represent the ___________and r is the _____________.

centerradius

How do you get the radius by itself?

Take the square root!

Ex 1: Graph the Equation of a Circle16)2()4( 22 yxGraph by finding the

center and radius of the circle.

Step 1: Identify the center & radius

222 )()( rkyhx

Center: ________ Radius: _________

(4, -2)4

Step 2: Plot the center and then 4 points to the left, right, up, and down.

Ex 1: Graph the Equation of a Circle

16)2()4( 22 yx

Let’s take it one step further…What if I want you to move the circle 3 units to the right

and 4units up? What would the equation be?Step 1: Write the original equation

(4, -2)Step 2: Determine the new center after the shift.

Center: _________

New Center: _____(7, 2)

Step 3: Write the new equation

16)2()7( 22 yx

You Try…Graph the following:

1. 2. 3622 xy25)3( 22 yx

Center: ________ Center: ________

Radius: _________ Radius: ________

(0, 3)

5

(0, 0)

6

You Try…Graph the following:

3. 4. 8)3( 22 yx10)2()3( 22 yx

Center: ________ Center: ________

Radius: _________ Radius: ________

(-3, 2)

≈ 3.162

(3, 0)

≈ 2.828

Let’s Try the Reverse…Can you write an equation of a circle given thecenter and radius? 222 )()( rkyhx

Example: Write an equation for a circle with center C(-3, 6) and a radius of 6 units. Graph it.

222 )()( rkyhx Step 1: Write the standard form of the equation

Step 2: Label h, k, and r h = -3 k = 6 r = 6

Step 3: Plug in your values and simplify!

222 6)6())3(( yx

36)6()3( 22 yx

You Try…Write the equation of the circle in standardform. Then, graph it!

1. Center: (0, 0) and 2. Center: (-3, 5) and radius of 5. diameter of 8. 2522 yx 16)5()3( 22 yx

PracticeWhat is an equation of the line tangent to thecircle at (-1, 3)?1022 yx

What is a tangent????

Remember, we learned in Geometry that a tangent to a circle is ______________ to the radius at a point of tangency.perpendicular

Step 1: Graph the circle

PracticeWhat is an equation of the line tangent to thecircle at (-1, 3)?

1022 yx

Step 2: Plot the point (-1,3) and determine

the slope of the radius. How will you do this?

31

3

run

risem

1022 yx

Step 3: What will the slope be of a line perpendicular to the radius?

3

1

TAKS PracticeWhat is an equation of the line tangent to thecircle at (-1, 3)?

1022 yx

1022 yxStep 4: Use point-slope form to find the equation using point (-1,3) and slope of 1/3.

)( 11 xxmyy

)1(3

13 xy

3

1

3

13 xy

3

10

3

1 xy

Completing the Square

1. Put in ax2 + bx + c = 0 form

2. Add/Subtract the c to the other side of the equation

3. If needed find the GCF (a has to be 1)

4. Half the b value and square it, and give that value to both sides of the equation.

5. Write the trinomial as a binomial squared.

Try using completing the Square

1. x2 – 6x – 4 = 0

2. x2 + 8x + 12 = 0

3. 3x2 + 12x – 5 = 0

Standard Form Center Form0118622 yxyx

1186 22 yyxx

Change into center form. Use completing the square!

Step 1: Write the equation with the number without the variable on the other side of the equal sign.

Step 2: Group your variables together if they are not already. (in this case, they are!)

1186 22 yyxx

Standard Form Center Form

1186 22 yyxx

92

62

162

82

3622 3x 4y

Center =

Radius =

4,3 6

Step 3: Complete the square for each one!

9 16 9 16

Step 4: Write each as a polynomial squared:

Step 5: Identify the center and radius and graph it!

Try these…1. What is an equation of the line tangent to the

circle at (-4, 7)?

2. Change into center form.1264 22 yyxx

25)3()1( 22 yx

25)3()2( 22 yx

1043 xy

In Class Assignment

Page 629

# 3, 5, 9, 11, 15, 17, 23, 25, 31 33

Homework

Worksheet