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Classifying Conics 10.6

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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

Can be multiplied out to look like this….

Can be written like this…..

Can be written like this…..

Can be written like this…..

- Pay close attention to whose squared and whose not…
- Look at the coefficients in front of the squared terms and their signs.

Both x and y are squared

And their coefficients are the same number and sign

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

- Either x or y is squared but not both

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

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

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 = CCircle

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

B2 − 4AC = 0Parabola

B2 − 4AC > 0Hyperbola

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

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

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 = CCircle

B2- 4AC < 0, B ≠ 0, A ≠ CEllipse

B2- 4AC = 0,Parabola

B2- 4AC > 0Hyperbola

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