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.
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