Section 8.1Conic Basics

Names of Conics Circle Ellipse Parabola Hyperbola

Definitions Circle

The set of all points, equidistant from the same point Parabola

The set of all points, equidistant from both a line and a point

Ellipse The set of all points, the sum of whose distances to two

fixed points is constant Hyperbola

The set of all points, the difference of whose distances to two fixed points is a constant

General form of a second degree equation in two variables

022 FEyDxCyBxyAx

Use the discriminant (B2 – 4AC) determine the shape of conics.

Conic DiscriminantEllipse B2 – 4AC < 0

Parabola B2 – 4AC = 0

Hyperbola B2 – 4AC > 0

Determine the type of conic42 22 yx

Step 1: Identify A,B,C,D,E, and F022 FEyDxCyBxyAx

A = 2 , B = 0, C = 1, D = 0, E = 0, F = -4

Step 2: Calculate the discriminant 12404 22 ACB

8 0

Ellipse

Note: If the equation is an ellipse, and A = C, then it is actually a circle.

Determine the type of conic42 22 yx

042 22 yx

A = 2, B = 0, C = -1, D = 0, E = 0, F = -4

12404 22 ACB

08

Hyperbola

Determine the type of Conic

12

2

2

2

b

y

a

x

012

2

2

2

b

y

a

x

1,0,0,1

,0,1

22 FED

bCB

aA

2222 11

404ba

ACB

22

4

ba0

Hyperbola