Writing equations of conics in vertex form MM3G2

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:

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.

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

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

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

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

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) 2 +9 𝑦+3 2 =36 Now we have to make the equation equal 1 and that will give us our denominators

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

Example 1 Step 7: simplify each fraction. 4(𝑥+1) 2 36 + 9 𝑦+3 2 36 = 36 36 (𝑥+1) 2 9 + 𝑦+3 2 4 =1 Now the equation looks like what we are used to 1 9 4

Example 2: Ellipse ( 4𝑥 2 −32𝑥 )+(25 𝑦 2 −150𝑦 )=−189 4𝑥 2 + 25𝑦 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

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−4 2 +25 𝑦−3 2 =100

Example 2 4 𝑥−4 2 +25 𝑦−3 2 =100 100 4(𝑥−4) 2 100 + 25 𝑦−3 2 100 = 100 100 (𝑥−4) 2 25 + 𝑦−3 2 4 =1 1 25 4

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

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+2 2 +4 𝑦−5 2 =324

Example 3 9 𝑥+2 2 +4 𝑦−5 2 =324 324 9(𝑥+2) 2 324 + 4 𝑦−5 2 324 = 324 324 (𝑥+2) 2 36 + 𝑦−5 2 81 =1 1 36 81

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

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

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) 2 18 − 9 𝑦+3 2 18 = 18 18 (𝑥+1) 2 18 − 𝑦+3 2 2 =1 1 2

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

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

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) 2 36 − 9 𝑥−4 2 36 = 36 36 (𝑦+2) 2 9 − 𝑥−4 2 4 =1 1 9 4