1. Writing equations of conics in vertex form
MM3G2

2. Recall: The equation for a circle does not have denominators
The equation for an ellipse and a hyperbola do have denominators
The equation for a circle is not equal to one
The equation for an ellipse and a hyperbola are equal to one
We have a different set of steps for converting ellipses and hyperbolas to the vertex form:

3. STEPS: Writing equations of ellipses and hyperbolas in vertex form
Step 1: move the constant to the other side of the equation and move common variables together
Step 2: Group the x terms together and the y terms together
Step 3: Factor the GCF (coefficient) from the x group and then from the y group
Step 4: Complete the square on the x group (don’t forget to multiply by the GCF before you add to the right side.)
Then do the same for the y terms.
Step 5: Write the factored form for the groups.

4. STEPS:
Step 6: Divide by the constant.
Step 7: simplify each fraction.

5. Write the equation for the ellipse in vertex form:
Example 1
4𝑥 2 + 9𝑦 2 +8𝑥+54𝑦+52=3
Step 1: move the constant to the other side of the equation and move common variables together
4𝑥 2 +8𝑥+ 9𝑦 2 +54𝑦=3−52
4𝑥 2 +8𝑥+ 9𝑦 2 +54𝑦=−49

6. Example 1 4(𝑥 2 +2𝑥 ) + 9(𝑦 2 +6𝑦 )=−49 4𝑥 2 +8𝑥+ 9𝑦 2 +54𝑦=−49
Step 2: Group the x terms together and the y terms together
(4𝑥 2 +8𝑥 )+ (9𝑦 2 +54𝑦 )=−49
Step 3: Factor the GCF (coefficient)from the x group
and then from the y group
4(𝑥 2 +2𝑥 )
+ 9(𝑦 2 +6𝑦 )=−49

7. Example 1 4 (𝑥 2 +2𝑥 +1)+ =−49+4 9 𝑦 2 +6𝑦 +9 +81
4(𝑥 2 +2𝑥 )+ 9(𝑦 2 +6𝑦 )=−49
Step 4: Complete the square on the x group (don’t forget to multiply by the GCF before you add to the right side.)
Then do the same for the y terms
2/2 = 1
6/2 = 3
12 = 1
32 = 9
4 (𝑥 2 +2𝑥 +1) =−49+4
9 𝑦 2 +6𝑦 +9
+81
4 (𝑥 2 +2𝑥 +1)+9 𝑦 2 +6𝑦 +9 =36

8. Example 1
4 (𝑥 2 +2𝑥 +1)+9 𝑦 2 +6𝑦 +9 =36
Step 5: Write the factored form for the groups.
4(𝑥+1) 𝑦+3 2 =36
Now we have to make the equation equal 1 and that will give us our denominators

9. Example 1 Step 6: Divide by the constant. 4(𝑥+1) 2 +9 𝑦+3 2 =36 36
4(𝑥+1) 𝑦+3 2 =36 36
4(𝑥+1) 𝑦 = 36 36

10. Example 1 Step 7: simplify each fraction.
4(𝑥+1) 𝑦 = 36 36
(𝑥+1) 𝑦 =1
Now the equation looks like what we are used to 1 9 4

11. Example 2: Ellipse ( 4𝑥 2 −32𝑥 )+(25 𝑦 2 −150𝑦 )=−189
4𝑥 𝑦 2 −32𝑥−150𝑦+189=0
4𝑥 2 −32𝑥+25 𝑦 2 −150𝑦=−189
( 4𝑥 2 −32𝑥 )+(25 𝑦 2 −150𝑦 )=−189
4( 𝑥 2 −8𝑥 )+25( 𝑦 2 −6𝑦 )=−189

12. Example 2 4( 𝑥 2 −8𝑥 )+25( 𝑦 2 −6𝑦 )=−189 4 (𝑥 2 −8𝑥 +16)+ =−189+64
4( 𝑥 2 −8𝑥 )+25( 𝑦 2 −6𝑦 )=−189
-8/2 = -4
-6/2 = -3
-42 = 16
-32 = 9
4 (𝑥 2 −8𝑥 +16) =−189+64
25 𝑦 2 −6𝑦 +9
+225
4 (𝑥 2 −8𝑥 +16)+25 𝑦 2 −6𝑦 +9 =100
4 𝑥− 𝑦−3 2 =100

13. Example 2
4 𝑥− 𝑦−3 2 =
4(𝑥−4) 𝑦− =
(𝑥−4) 𝑦− =1 1 25 4

14. Example 3: Ellipse ( 9𝑥 2 +36𝑥 )+(4 𝑦 2 −40𝑦 )=188
9𝑥 2 + 4𝑦 2 +36𝑥−40𝑦−100=88
9𝑥 2 +36𝑥+4 𝑦 2 −40𝑦=188
( 9𝑥 2 +36𝑥 )+(4 𝑦 2 −40𝑦 )=188
9( 𝑥 2 +4𝑥 )+4( 𝑦 2 −10𝑦 )=188

15. Example 3 9( 𝑥 2 +4𝑥 )+4( 𝑦 2 −10𝑦 )=188 9 (𝑥 2 +4𝑥 +4)+ =188+36
9( 𝑥 2 +4𝑥 )+4( 𝑦 2 −10𝑦 )=188
4/2 = 2
-10/2 = -5
22 = 4
-52 = 25
9 (𝑥 2 +4𝑥 +4) =188+36
4 𝑦 2 −10𝑦 +25
+100
9 (𝑥 2 +4𝑥 +4)+4 𝑦 2 −10𝑦 +25 =324
9 𝑥 𝑦−5 2 =324

16. Example 3
9 𝑥 𝑦−5 2 =
9(𝑥+2) 𝑦− =
(𝑥+2) 𝑦− =1 1 36 81

17. Example 4: Hyperbola 𝑥 2 +2𝑥 + −9 𝑦 2 −54𝑦 =98
𝑥 2 −9 𝑦 2 +2𝑥−54𝑦−98=0
𝑥 2 +2𝑥−9 𝑦 2 −54𝑦=98
𝑥 2 +2𝑥 −9 𝑦 2 −54𝑦 =98
𝑥 2 +2𝑥 −9( 𝑦 2 +6𝑦 )=98

18. Example 4 𝑥 2 +2𝑥 −9( 𝑦 2 +6𝑦 )=98 (𝑥 2 +2𝑥 +1) =98+1 −9 𝑦 2 +6𝑦 +9
𝑥 2 +2𝑥 −9( 𝑦 2 +6𝑦 )=98
2/2 = 1
6/2 = 3
12 = 1
32 = 9
(𝑥 2 +2𝑥 +1) =98+1
−9 𝑦 2 +6𝑦 +9
−81
(𝑥 2 +2𝑥 +1)−9 𝑦 2 +6𝑦 +9 =18
𝑥+1 2 −9 𝑦+3 2 =18

19. Example 4 𝑥+1 2 −9 𝑦+3 2 =18 18 (𝑥+1) 2 18 − 9 𝑦+3 2 18 = 18 18
𝑥+1 2 −9 𝑦+3 2 =18 18
(𝑥+1) − 9 𝑦 = 18 18
(𝑥+1) − 𝑦 =1 1 2

20. Example 5: Hyperbola 4𝑦 2 +16𝑦 + −9 𝑥 2 +72𝑥 =164
4𝑦 2 −9 𝑥 2 +16𝑦+72𝑥−164=0
4𝑦 2 +16𝑦−9 𝑥 2 +72𝑥=164
4𝑦 2 +16𝑦 −9 𝑥 2 +72𝑥 =164
4 𝑦 2 +4𝑦 −9( 𝑥 2 −8𝑥 )=164

21. Example 5 4 𝑦 2 +4𝑦 −9( 𝑥 2 −8𝑥 )=164 4(𝑦 2 +4𝑦 +4) =164+16
4 𝑦 2 +4𝑦 −9( 𝑥 2 −8𝑥 )=164
4/2 = 2
-8/2 = -4
22 = 4
-42 =16
4(𝑦 2 +4𝑦 +4) =164+16
−9 𝑥 2 −8𝑥 +16
−144
4(𝑦 2 +4𝑦 +4)−9 𝑥 2 −8𝑥 +16 =36
4 𝑦+2 2 −9 𝑥−4 2 =36

22. Example 5 4 𝑦+2 2 −9 𝑥−4 2 =36 36 4(𝑦+2) 2 36 − 9 𝑥−4 2 36 = 36 36
4 𝑦+2 2 −9 𝑥−4 2 =36 36
4(𝑦+2) − 9 𝑥− = 36 36
(𝑦+2) 2 9 − 𝑥− =1 1 9 4