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### The equation of any conic can be written in the form-

Classifying Conics10.6

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

What information can the discriminant tell you about a conic?

Called a general 2nd degree equation

Circles

Can be multiplied out to look like this….

Ellipse

Can be written like this…..

Parabola

Can be written like this…..

Hyperbola

Can be written like this…..

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.

Ellipse

- Both x and y are squared
- Their coefficients are different but their signs remain the same.

Parabola

- Either x or y is squared but not both

Hyperbola

- Both x and y are squared
- Their coefficients are different and so are their signs.

You Try!

- Ellipse
- Parabola
- Hyperbola
- Circle
- Hyperbola
- Parabola
- Circle
- Ellipse
- Hyperbola
- 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 Conic

2x2 + 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)

Complete the Square

Steps to Complete the Square

1. 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

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 = C Circle

B2- 4AC < 0, B ≠ 0, A ≠ C Ellipse

B2- 4AC = 0, Parabola

B2- 4AC > 0 Hyperbola

Assignment 10.6

Page 628, 29-55 odd

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