1. MTH253 Calculus III Chapter 10, Part I (sections 10.1 – 10.3) Conic Sections

2. Conics The intersection of a right circular cone and a plane will produce one of four curves: Parabola Ellipse Circle Hyperbola

3. Conics Not rotated ◦ B = 0 ◦ Lines of symmetry are horizontal or vertical Not translated ◦ Parabola: C, D, & F = 0 or A, E, & F = 0  Vertex at the origin ◦ Ellipse & Hyperbola: D = 0 & E = 0  “Centered” at the origin

4. The Parabola Vertical Axis of Symmetry (0, p) (0,0) (2p,p) Note: If p < 0, then just flip this upside-down. i.e. the y-axis

5. The Parabola - Example

6. Find the equation of the parabola with its vertex at the origin and directrix the equation y = –5

7. The Parabola Horizontal Axis of Symmetry (p,0) (0,0) (p,2p) Note: If p < 0, then just flip this to the right. i.e. the x-axis

8. The Ellipse Horizontal Major Axis (c,0) (0,b) (a,0)

9. The Ellipse - Example

10. Find the equation of the ellipse with its foci at (  2,0) and eccentricity of 0.25.

11. The Ellipse Vertical Major Axis (0,c) (0,a) (b,0)

12. The Circle (0,r) (r,0) (0,0)

13. The Hyperbola Horizontal Focal Axis (c,0) (0,b) (a,0)

14. The Hyperbola - Example

15. Find the equation of the hyperbola with its vertices at (  3,0) and a directrix x = 2.

16. The Hyperbola Vertical Focal Axis (0,c) (b,0) (0,a)

17. PF = e * PD F P D F P D D P F e < 1e > 1 e = 1

18. Translations To move the center of an ellipse or hyperbola or the vertex of a parabola to the point (h, k), replace x with x-h and y with y-k. Treat (h, k) as if it was the origin.

19. Translations – “Complete the Square” Examples:

20. Rotations – The “Cross Product Term” Angle of Rotation Substitutions Note: Use the rotations calculator!

21. The Descriminant