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

2. 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.

3. Conic Sections: Eccentricity
For both an ellipse and a hyperbola
where c is the distance from the center to the focus and a is the distance from the center to a vertex.

4. Classifying Conics 10.6
What is the general 2nd degree equation for any conic?
What information can the discriminant tell you about a conic?

5. The equation of any conic can be written in the form-
Called a general 2nd degree equation

6. Circles
Can be multiplied out to look like this….

7. Ellipse
Can be written like this…..

8. Parabola
Can be written like this…..

9. Hyperbola
Can be written like this…..

10. 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.

11. Circle Both x and y are squared
And their coefficients are the same number and sign

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

13. Parabola
Either x or y is squared but not both

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

15. Ellipse
Parabola
Hyperbola
Circle
You Try!

16. 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 = Parabola
B2 − 4AC > Hyperbola

17. Classify the Conic 2x2 + y2 −4x − 4 = 0 Ax2 +Bxy +Cy2 +Dx +Ey +F = 0
B2 − 4AC = 02 − 4(2)(1) = −8
B2 − 4AC < 0, the conic is an ellipse

18. Graph the Conic 2x2 + y2 −4x − 4 = 0 2x2 −4x + y2 = 4
V(1±√6), CV(1±√3)
Complete the Square

19. 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

20. Write the equation in standard form by completing the square

21. 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

22. Assignment 10.6
Page 628, odd