To each conic section (ellipse, parabola, hyperbola, circle) there is a number called the eccentricity that uniquely characterizes the shape of the curve.

Conic Sections: Eccentricity

If e = 1, the conic is a parabola.

If e = 0, the conic is a circle.

If e < 1, the conic is an ellipse.

If e > 1, the conic is a hyperbola.

Conic Sections: Eccentricity

For both an ellipse and a hyperbola

eca

where c is the distance from the center to the focus and a is the distance from the center to a vertex.

Conic Sections: Eccentricity

Classifying Conics10.6

What is the general 2nd degree equation for any conic?

What information can the discriminant tell you about a conic?

The equation of any conic can be written

in the form-

Called a general 2nd degree equation

2 2 0Ax Bxy Cy Dx Ey F

Circles

Can be multiplied out to look like this….

2 2( 1) ( 2) 16x y

2 2 2 4 11 0x y x y

Ellipse

Can be written like this…..

22( 1)

( 1) 14

xy

2 24 2 8 1 0x y x y

Parabola

Can be written like this…..

2( 6) 4( 8)y x

2 12 4 4 0y y x

Hyperbola

Can be written like this…..

22 ( 4)

( 4) 19

yx

2 29 72 8 1 0x y x y

How do you know which conic it is when it’s been multiplied

out?

• Pay close attention to whose squared and whose not…

• Look at the coefficients in front of the squared terms and their signs.

Circle Both x and y are

squared

And their coefficients are the same number and sign

2 2 2 4 11 0x y x y

Ellipse• Both x and y are

squared• Their coefficients are

different but their signs remain the same.

2 24 2 8 1 0x y x y

Parabola• Either x or y is

squared but not both

2 12 4 4 0y y x

Hyperbola

• Both x and y are squared

• Their coefficients are different and so are their signs.

2 29 72 8 1 0x y x y

You Try!

0343.10

036164.9

0782.8

0593033.7

0164y8x2x6.

046y6x2y2x.5

0314y12x2y2x4.

03023y25x3.

041y20x22x2.

032x24y2x1.

22

22

22

22

yyx

yxyx

xyx

xyx

1.Ellipse

2.Parabola

3.Hyperbola

4.Circle

5.Hyperbola

6.Parabola

7.Circle

8.Ellipse

9.Hyperbola

10.Ellipse

When you want to be sure…

of a conic equation, then find the type of conic using discriminate information:

Ax2 +Bxy +Cy2 +Dx +Ey +F = 0

B2 − 4AC < 0, B = 0 & A = C Circle

B2 − 4AC < 0 & either B≠0 or A≠C Ellipse

B2 − 4AC = 0 Parabola

B2 − 4AC > 0 Hyperbola

Classify the Conic

2x2 + y2 −4x − 4 = 0

Ax2 +Bxy +Cy2 +Dx +Ey +F = 0

A = 2

B = 0

C = 1

B2 − 4AC = 02 − 4(2)(1) = −8

B2 − 4AC < 0, the conic is an ellipse

Graph the Conic2x2 + y2 −4x − 4 = 0

2x2 −4x + y2 = 4

2(x2 −2x +___)+ y2 = 4 + ___ (−2/2)2= 1

2(x2 −2x +1)+ y2 = 4 + 2(1)

2(x−1)2 + y2 = 6

V(1±√6), CV(1±√3)

166

)1(2 22

yx

163

)1( 22

yx

Complete the Square

Steps to Complete the Square1. Group x’s and y’s. (Boys with the boys and

girls with the girls) Send constant numbers to the other side of the equal sign.

2. The coefficient of the x2 and y2 must be 1. If not, factor out.

3. Take the number before the x, divide by 2 and square. Do the same with the number before y.

4. Add these numbers to both sides of the equation. *(Multiply it by the common factor in #2)

5. Factor

Write the equation in standard form by completing the square

01824 22 yxyx

______1___)2(4___2 22 yyxx

1)2(42 22 yyxx

)1)(4(11)12(4)12( 22 yyxx

4)1(4)1(22 yx

4

4

4

)1(4

4

)1( 22

yx

11

)1(

4

)1( 22

yx

12

22

What is the general 2nd degree equation for any conic?

What information can the discriminant tell you about a conic?

B2- 4AC < 0, B = 0, A = CCircle

B2- 4AC < 0, B ≠ 0, A ≠ CEllipse

B2- 4AC = 0, Parabola

B2- 4AC > 0 Hyperbola

2 2 0Ax Bxy Cy Dx Ey F

Assignment 10.6

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