Algebra 2 Notes Name: ________________ Section 10.6 – Identifying Conic Sections

The conic sections result from intersecting a plane with

a double cone, as shown in the figure. There are three

distinct families of conic sections:

- the ellipse (including the circle)

- the parabola (with one branch)

- the hyperbola (with two branches)

Before we work in detail with these various conic

sections, we are going to practice identifying them. ☺

Standard Form of Conic Sections:

Conic Equation in Standard Form

CIRCLE ( ) ( )2 2 2x h y k r− + − =

Horizontal Axis Vertical Axis

ELLIPSE ( ) ( )2 2

2 21

x h y k

a b

− −+ =

( ) ( )2 2

2 21

x h y k

b a

− −+ =

HYBERBOLA ( ) ( )2 2

2 21

x h y k

a b

− −− =

( ) ( )2 2

2 21

y k x h

a b

− −− =

PARABOLA ( ) ( )24y k p x h− = −

( ) ( )24x h p y k− = −

Example 1: Identify the conic section that each equation represents.

a. 2 2

2 21

7 8

x y+ =

b. ( )2 8 2y x= + c.

2 2

2 2

( 2) ( 3)1

5 2

x y− −− =

d. ( ) ( )2 2 21 2 3x y+ + − =

e. ( ) ( )

2 23 1

14 9

y x+ −− =

f. ( ) ( )2

5 4 2x y+ = − −

You will not always be given the equation of a conic section in standard form, so you need to be able to

identify the conic section based on some generalizations regarding their equations. The following table will

help you determine the type of conic section if you have the equation set equal to zero.

Conic Example Generalization

CIRCLE 2 24 4 3 2 12 0x y x y+ − + − =

ELLIPSE 2 25 4 2 0x y x+ − =

OR 2 23 10 10 0x y x y+ − + − =

HYBERBOLA 2 2 2 4 2 0x y x y− + − − =

OR 2 24 7 3 2 15 0y x x y− − − − =

PARABOLA 2 3 4 1 0x x y− + + =

OR 2 5 6 0y x+ − =

Example 2: Identify each conic section. Justify your answer.

a. 2 24 9 18 27 0x y y− − + =

b. 2 2 4 6 36 0x y x y+ − + − =

c. 2 23 4 6 4 0x y x y+ + + + =

d. 2 2 3 8 0y y x+ − − =

e. ( )2 22 4 36x y+ =

f. 2 23 6 10 3 2x x y y+ + = −

g. 2 2 22 4 6 4 2x y x y y+ − − + =

h. 2 2 25 5 2 25 0x y x+ − − =