WRITING EQUATIONS OF CONICS IN VERTEX FORMMM3G2
Write the equation for the circle in vertex form:
Example 1
Step 1: Move the constant to the other side of the equation & put your common variables together
Example 1
Step 2: Identify the coefficients of the squared terms and divide everything by that coefficient. Both coefficients are 1 so divide
everything by 1
Example 1
Step 3: Group the x terms together and the y terms together using parenthesis.
Example 1
Step 4: Complete the square for the x terms
Then for the y terms
22=1 12=1 −42 =−2 (−2)2=4
Example 1
Step 5: Write the factored form for the groups.
What is the center of this circle?
What is the radius?
Write the equation for the circle in vertex form:
Example 2
Step 1: Move the constant to the other side of the equation & put your common variables together
Example 2
Step 2: Identify the coefficients of the squared terms and divide everything by that coefficient. Both coefficients are 2 so divide
everything by 2
Example 2
Step 3: Group the x terms together and the y terms together using parenthesis.
Example 2
Step 4: Complete the square for the x terms
Then for the y terms
62=3 32=9 42=2 22=4
Example 2
Step 5: Write the factored form for the groups.
What is the center of this circle?
What is the radius?
Write the equation for the circle in vertex form:
Example 3
Step 1: Move the constant to the other side of the equation & put your common variables together
Example 3
Step 2: Identify the coefficients of the squared terms and divide everything by that coefficient. Both coefficients are 4 so divide
everything by 4
Example 3
Step 3: Group the x terms together and the y terms together using parenthesis.
Example 3
Step 4: Complete the square for the x terms
Then for the y terms
62=3 32=9 82=4 42=16
Example 3
Step 5: Write the factored form for the groups.
What is the center of this circle?
What is the radius?
Write the equation for the circle in vertex form:
Example 4
Step 1: Move the constant to the other side of the equation & put your common variables together
Example 4
Step 2: Identify the coefficients of the squared terms and divide everything by that coefficient. Both coefficients are 5 so divide
everything by 5
Example 4
Step 3: Group the x terms together and the y terms together using parenthesis.
Example 4
Step 4: Complete the square for the x terms
Then for the y terms
−162 =−8¿ 4
2=2 22=4
Example 4
Step 5: Write the factored form for the groups.
What is the center of this circle?
What is the radius?
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:
Write the equation for the ellipse in vertex form:
Example 5
Step 1: Move the constant to the other side of the equation and move common variables together
Example 5
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
Example 5
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
22=1 12=1
62=3 32=9
4(𝑥¿¿2+2𝑥+1)+9 ( 𝑦2+6 𝑦+9 )=36¿
9 ( 𝑦2+6 𝑦+9 ) +81
Example 5
Step 5: Write the factored form for the groups.
**Now we have to make the equation equal 1 and that will give us our denominators
Example 5 Step 6: Divide by the constant.
Example 5 Step 7: simplify each fraction.
Now the equation looks like what we are used to!!
9 41
(𝑥+1)2
9+
(𝑦+3 )2
4=1
What is the center of this ellipse?
What is the length of the major axis?
What is the length of the minor axis?
Example 6: Ellipse
Step 2:
Step 1:
Step 3:
Example 6
− 82=−4 −42=16 − 62=−3 −3 2=9
4(𝑥¿¿2−8𝑥+16)+25 (𝑦 2−6 𝑦+9 )=100¿
25 ( 𝑦2−6 𝑦+9 ) +225
4 (𝑥−4 )2+25 (𝑦−3 )2=100
Step 4:
Step 5:
Example 6
25
41
Step 6:
(𝑥−4)2
25+
(𝑦−3 )2
4=1
What is the center of this ellipse?
What is the length of the major axis?
What is the length of the minor axis?
Example 7: Ellipse
Step 2:
Step 1:
Step 3:
Example 7
42=2 22=4−
102 =−5 −52=25
9 (𝑥¿¿ 2+4 𝑥+4)+4 ( 𝑦2−10 𝑦+25 )=324 ¿
4 ( 𝑦2−10 𝑦+25 ) +100
9 (𝑥+2 )2+4 (𝑦−5 )2=324
Step 4:
Step 5:
Example 7
36
811
Step 6:
(𝑥+2)2
36+
(𝑦−5 )2
81=1
What is the center of this ellipse?
What is the length of the major axis?
What is the length of the minor axis?
Example 8: Hyperbola
Step 2:
Step 1:
Step 3:
Example 8
22=1 12=1
62=3 32=9
(𝑥¿¿2+2 𝑥+1)−9 ( 𝑦2+6 𝑦+9 )=18 ¿
−9 (𝑦2+6 𝑦+9 ) −81
(𝑥+1 )2−9 (𝑦+3 )2=18
Step 4:
Step 6:
Example 8
21
Step 6:
(𝑥+1)2
18− (𝑦+3 )2
2=1
What is the center of this hyperbola?
What is the length of the transverse axis?
What is the length of the conjugate axis?
Example 9: Hyperbola
Step 2:
Step 1:
Step 3:
Example 9
42=2 22=4 − 82=−4−42=16
4(𝑦¿¿ 2+4 𝑦+4)−9 (𝑥2−8 𝑥+16 )=36¿
−9 (𝑥2−8 𝑥+16 ) −144
4 (𝑦+2 )2−9 (𝑥−4 )2=36
Step 4:
Step 5:
Example 9
9 41
Step 6:
(𝑦+2)2
9− (𝑥−4 )2
4=1
What is the center of this hyperbola?
What is the length of the transverse axis?
What is the length of the conjugate axis?
You Try! Write the equation of each conic section
in vertex form:
Identify the center of each conic section as well as the length of the major/minor or
transverse/conjugate axis.