1. Writing equations of circles in vertex form
MM3G2

2. Write the equation for the circle in vertex form:
Example 1
𝑥 2 + 𝑦 2 +2𝑥−4𝑦−4=0
Step 1: Move the constant to the other side of the equation & put your common variables together
𝑥 2 +2𝑥+ 𝑦 2 −4𝑦−4+4=0+4
𝑥 2 +2𝑥+ 𝑦 2 −4𝑦=4

3. Example 1
𝑥 2 +2𝑥+ 𝑦 2 −4𝑦=4
Step 2: Identify the coefficients of the squared terms and divide everything by that coefficient.
Both coefficients are 1 so divide everything by 1
𝑥 2 +2𝑥+ 𝑦 2 −4𝑦=4 1

4. Example 1
𝑥 2 +2𝑥+ 𝑦 2 −4𝑦=4
Step 3: Group the x terms together and the y terms together using parenthesis.
(𝑥 2 + 2𝑥 )+ 𝑦 2 −4𝑦 =4

5. Example 1 Step 4: Complete the square for the x terms
(𝑥 2 + 2𝑥 )+ 𝑦 2 −4𝑦 =4
Step 4: Complete the square for the x terms
Then for the y terms
(𝑥 2 +2𝑥 +1)+ 𝑦 2 −4𝑦 +4 =4+1+4
(𝑥 2 + 2𝑥 +1)+ 𝑦 2 −4𝑦 +4 =9
2 2 =1
−4 2 =−2
12 =1
(−2)2 = 4

6. What is the center of this circle?
Example 1
(𝑥 2 + 2𝑥 +1)+ 𝑦 2 −4𝑦 +4 =9
Step 5: Write the factored form for the groups.
(𝑥+1) 2 + 𝑦−2 2 =9
What is the center of this circle?
(−1, 2)
What is the radius?
𝑟=3

7. Write the equation for the circle in vertex form:
Example 2
2𝑥 2 + 2𝑦 2 +12𝑥+8𝑦+4=28
Step 1: Move the constant to the other side of the equation & put your common variables together
2𝑥 2 +12𝑥+ 2𝑦 2 +8𝑦+4−4=28−4
2𝑥 2 +12𝑥+ 2𝑦 2 +8𝑦=24

8. Example 2 𝑥 2 +6𝑥+ 𝑦 2 +4𝑦=12 2𝑥 2 +12𝑥+ 2𝑦 2 +8𝑦=24
Step 2: Identify the coefficients of the squared terms and divide everything by that coefficient.
Both coefficients are 2 so divide everything by 2
2𝑥 2 +12𝑥+ 2𝑦 2 +8𝑦=24 2
𝑥 2 +6𝑥+ 𝑦 2 +4𝑦=12

9. Example 2
𝑥 2 +6𝑥+ 𝑦 2 +4𝑦=12
Step 3: Group the x terms together and the y terms together using parenthesis.
(𝑥 2 +6𝑥 )+ 𝑦 2 +4𝑦 =12

10. Example 2 Step 4: Complete the square for the x terms
(𝑥 2 +6𝑥 )+ 𝑦 2 +4𝑦 =12
Step 4: Complete the square for the x terms
Then for the y terms
(𝑥 2 +6𝑥 +9)+ 𝑦 2 +4𝑦 +4 =12+9+4
(𝑥 2 +6𝑥 +9)+ 𝑦 2 +4𝑦 +4 =25
6 2 = 3
4 2 = 2
32 = 9
22 = 4

11. What is the center of this circle?
Example 2
(𝑥 2 +6𝑥+9)+ 𝑦 2 +4𝑦 +4 =25
Step 5: Write the factored form for the groups.
(𝑥+3) 2 + 𝑦+2 2 =25
What is the center of this circle?
(−3, −2)
What is the radius?
𝑟=5

12. Write the equation for the circle in vertex form:
Example 3
4𝑥 2 +4 𝑦 2 +24𝑥+32𝑦+13=157
Step 1: Move the constant to the other side of the equation & put your common variables together
4𝑥 2 +24𝑥+4 𝑦 2 +32𝑦+13−13=157−13
4𝑥 2 +24𝑥+4 𝑦 2 +32𝑦=144

13. Example 3
4𝑥 2 +24𝑥+4 𝑦 2 +32𝑦=144
Step 2: Identify the coefficients of the squared terms and divide everything by that coefficient.
Both coefficients are 4 so divide everything by 4
4𝑥 2 +24𝑥+4 𝑦 2 +32𝑦=144 4
𝑥 2 +6𝑥+ 𝑦 2 +8𝑦=36

14. Example 3
𝑥 2 +6𝑥+ 𝑦 2 +8𝑦=36
Step 3: Group the x terms together and the y terms together using parenthesis.
( 𝑥 2 +6𝑥 )+( 𝑦 2 +8𝑦 )=36

15. Example 3 Step 4: Complete the square for the x terms
( 𝑥 2 +6𝑥 )+( 𝑦 2 +8𝑦 )=36
Step 4: Complete the square for the x terms
Then for the y terms
(𝑥 2 +6𝑥 +9)+ 𝑦 2 +8𝑦 +16 =
𝑥 2 +6𝑥+9 + 𝑦 2 +8𝑦+16 =61
6 2 = 3
8 2 =4
32 = 9
42 =16

16. What is the center of this circle?
Example 3
𝑥 2 +6𝑥+9 + 𝑦 2 +8𝑦+16 =61
Step 5: Write the factored form for the groups.
𝑥 𝑦+4 2 =61
What is the center of this circle?
(−3, 4)
What is the radius?
𝑟= 61

17. Write the equation for the circle in vertex form:
Example 4
5𝑥 2 +5 𝑦 2 −80𝑥+20𝑦−34=106
Step 1: Move the constant to the other side of the equation & put your common variables together
5𝑥 2 −80𝑥+5 𝑦 2 +20𝑦−34+34=106+34
5𝑥 2 −80𝑥+5 𝑦 2 +20𝑦=140

18. Example 4
5𝑥 2 −80𝑥+5 𝑦 2 +20𝑦=140
Step 2: Identify the coefficients of the squared terms and divide everything by that coefficient.
Both coefficients are 5 so divide everything by 5
5𝑥 2 −80𝑥+5 𝑦 2 +20𝑦=140 5
𝑥 2 −16𝑥+ 𝑦 2 +4𝑦=28

19. Example 4
𝑥 2 −16𝑥+ 𝑦 2 +4𝑦=28
Step 3: Group the x terms together and the y terms together using parenthesis.
( 𝑥 2 −16𝑥 )+( 𝑦 2 +4𝑦 )=28

20. Example 4 Step 4: Complete the square for the x terms
(𝑥 2 − 16𝑥 )+ 𝑦 2 +4𝑦 =28
Step 4: Complete the square for the x terms
Then for the y terms
(𝑥 2 −16𝑥+64)+ 𝑦 2 +4𝑦 +4 =
𝑥 2 −16𝑥 𝑦 2 +4𝑦+4 =96
−16 2 =−8
4 2 = 2
(− 8) 2 =64
22 = 4

21. What is the center of this circle?
Example 4
𝑥 2 −16𝑥 𝑦 2 +4𝑦+4 =96
Step 5: Write the factored form for the groups.
𝑥− 𝑦+2 2 =96
What is the center of this circle?
(8,−2)
What is the radius?
𝑟= 96

22. What is the center of this circle?
You Try!
Write the following equation of a circle in vertex form:
4𝑥 2 +4 𝑦 2 +8𝑥−24𝑦+4=0
What is the center of this circle?
What is the radius?