1. Circles
Students will be able to transform an equation of a circle in standard form to center, radius form
by using the complete the square method.

2. Circles – Warm Up Simplify. 1. 16 2. 49 3. 20 4. 48 5. 72
Find the missing value to complete the square.
6. x2 – 2x x x x2 – 6x +
Find the missing value to complete the square.
6. x2 – 2x x x x2 – 6x +

3. Solutions
6. x2 – 2x + ; c = = – = (–1)2 = 1
7. x x + ; c = = = = 4
8. x2 – 6x + ; c = = – = (–3)2 = 9 b 2 4 6
= 4
= 7
=  5 =
=  3 =
=  2 =

4. CIRCLE TERMS EQUATION FORM CENTER RADIUS MIDPOINT FORMULA DISTANCE
Definition: A circle is an infinite number of points a set distance away from a center
EQUATION FORM
CENTER
RADIUS
MIDPOINT
FORMULA
DISTANCE r
(x – h)² + (y – k)² = r²
(h, k )
C=(h , k) r

5. Circles
Write an equation of a circle with center (3, –2) and radius 3.
(x – h)2 + (y – k)2 = r2 Use the standard form of the equation of a circle.
(x – 3)2 + (y – (–2))2 = 32 Substitute 3 for h, –2 for k, and
3 for r.
(x – 3)2 + (y + 2)2 = 9 Simplify.
Check: Solve the equation for y and enter both functions into your graphing calculator.
(x – 3)2 + (y + 2)2 = 9
(y + 2)2 = 9 – (x – 3)2
y = ± – (x – 3)2
y = –2 ± – (x – 3)2

6. Circles
Write an equation for the translation of x2 + y2 = 16 two units right and one unit down.
(x – h)2 + (y – k)2 = r 2 Use the standard form of the equation of a circle.
(x – 2)2 + (y – (–1))2 = 16 Substitute 2 for h, –1 for k, and 16 for r 2.
(x – 2)2 + (y + 1)2 = 16 Simplify.
The equation is (x – 2)2 + (y + 1)2 = 16.

7. Circles
Write an equation for the translation of x2 + y2 = 16 two units right and one unit down.
(x – h)2 + (y – k)2 = r 2 the equation of a circle.
(x – 2)2 + (y + 1)2 = 16
The equation is (x – 2)2 + (y + 1)2 = 16.
Now multiply it out for standard form:

8. Circles
Write an equation for the translation of x2 + y2 = 16 two units right and one unit down.
(x – h)2 + (y – k)2 = r 2 the equation of a circle.
(x – 2)2 + (y + 1)2 = 16

9. WRITE the equation of a circle in center, radius form
Group the x and y terms
Move the constant term
Complete the square for x, y
Write each variable as a square
Ex: x² + y² - 4x + 8y + 11 = 0

10. WRITE and GRAPH x² + y² - 4x + 8y + 11 = 0 Group the x and y terms
Given the equation of the circle in standard form
x² + y² - 4x + 8y + 11 = 0
Group the x and y terms
x² - 4x + y² + 8y + 11 = 0
Complete the square for x, y
x² - 4x y² + 8y + 16 =
(x – 2)² + (y + 4)² = 9
B) GRAPH
Plot Center (2,-4)
Radius = 3

11. WRITE and then GRAPH 4x² + 4y² + 36y + 5 = 0
Group terms, move constant, factor out the coefficient.
Complete the square for x,y
Divide both sides by coefficient

12. WRITE and then GRAPH This time factor out the coefficients:
4x² + 4y² + 36y + 5 = 0
Group terms, move constant, factor out the coefficient.
4(x²) + 4(y² + 9y + __ ) = -5 + ___
Notice that x is already done!

13. WRITE and GRAPH 4x² + 4y² + 36y + 5 = 0 Complete the square for y:
A) write the equation of the circle in standard form
4x² + 4y² + 36y + 5 = 0
Complete the square for y:
4(x²) + 4(y² + 9y+__) = -5 +__
4(x²)+4(y² + 9y + 81/4) =-5+81
4(x²) + 4(y + 9/2)² = 76
Divide by 4:
x² + (y + 9/2)² = 19
Center (0 , -9/2)
Radius =

14. WRITING EQUATIONS Write the EQ of a circle that has a center of (-5,7) and passes through (7,3)
Plot your info
Need to find values for h, k, r
(h , k) = ( , )
How do we find r?
Use distance form. for C and P.
Plug into circle formula
(x – h)² + (y – k)² = r²
C = (-5,7)
P = (7,3)
radius

15. WRITING EQUATIONS Write the EQ of a circle that has a center of (-5,7) and passes through (7,3)
Plot your info
Need to find values for h, k, r
(h , k) = (-5 , 7)
How do we find r?
Use distance form. for C and P.
Plug into circle formula
(x – h)² + (y – k)² = r²
(x + 5)² + (y – 7)² = ²
(x + 5)² + (y – 7)² = 160
C = (-5,7)
P = (7,3)
radius

16. Let’s Try One Write the EQ of a circle that has endpoints of the diameter at (-4,2) and passes through (4,-6)
Plot your info
Need to find values for h, k, r
How do we find (h,k)?
Use midpoint formula
(h , k) =
How do we find r?
Use dist form with C and B.
Plug into formula
(x – h)² + (y – k)² = r²
A = (-4,2)
radius
B = (4,-6)
Hint: Where is the center? How do you find it?

17. Let’s Try One Write the EQ of a circle that has endpoints of the diameter at (-4,2) and passes through (4,-6)
Plot your info
Need to find values for h, k, r
How do we find (h,k)?
Use midpoint formula
(h , k) = (0 , -2)
How do we find r?
Use dist form with C and B.
Plug into formula
(x – h)² + (y – k)² = r²
(x)² + (y + 2)² = 32
A = (-4,2)
radius
B = (4,-6)
Hint: Where is the center? How do you find it?

18. Suppose the equation of a circle is (x – 5)² + (y + 2)² = 9
Write the equation of the new circle given that:
A) The center of the circle moved up 4 spots and left 5:
(x – 0)² + (y – 2)² = 9
Center moved from (5,-2)  (0,2)
B) The center of the circle moved down 3 spots and right 6:
(x – 11)² + (y + 5)² = 9
Center moved from (5,-2)  (11,-5)

19. Let‘s Try One
Find the center and radius of the circle with equation (x + 4)2 + (y – 2)2 = 36.
(x – h)2 + (y – k)2 = r 2 Use the standard form.
(x + 4)2 + (y – 2)2 = 36 Write the equation.
(x – (–4))2 + (y – 2)2 = 62 Rewrite the equation in standard form.
h = –4 k = r = 6 Find h, k, and r.
The center of the circle is (–4, 2). The radius is 6.

20. Let’s Try One Graph (x – 3)2 + (y + 1)2 = 4.
(x – h)2 + (y – k)2 = r 2 Find the center and radius of the circle.
(x – 3)2 + (y – (–1))2 = 4
h = k = – r 2 = 4, or r = 2
Draw the center (3, –1) and radius 2.
Draw a smooth curve.

21. Solving linear quadratic systems Do now: page 182 # 37

22. Solving a circle-line system
Plot the line b=-2 m = -1
Plot the center and count in four directions for the radius
Find the points of intersection
y=-x -2
x² + (y + 2)² = 32
Y=-x-2

23. Solving a circle-line system
Plot the line b=-2 m = -1
Plot the center and count in four directions for the radius
Find the points of intersection
y=-x -2
x² + (y + 2)² = 32
(-4,2)
Y=-x-2
(4,-6)

24. Solving a circle-line system
By algebraic method: substitute –x-2 for y into second equation
y=-x -2
x² + (y + 2)² = 32

25. Solving a circle-line system
By algebraic method: substitute –x-2 for y into second equation
y=-x -2
x² + (y + 2)² = 32
x² + (-x )² = 32
x² + ( -x)² = 32
2x2=32
X2=16
X=4, x=-4 solve for y:
Y=-4-2 y=4-2
Y=-6 (4,-6) y=2 (-4, 2)

26. Lets solve this one both ways

27. Lets solve this one both ways