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

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

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

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

The Parabola - Example

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

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

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

The Ellipse - Example

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

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

The Circle (0,r) (r,0) (0,0)

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

The Hyperbola - Example

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

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

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

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.

Translations – “Complete the Square” Examples:

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

The Descriminant