ConicsWritten by Gaurav Rao

Last edited: 10/3/15

What Are Conics?• Conics are cross sections of a cone

• Locus of points that distance from a point (focus) And a line (directrix) are at a fixed ratio(eccentricity)

Parts of conic

• All conics have a Focus, and all conics but circles have diretricies• The eccentricity, defined as the ratio of the distances between the

focus and the directrix, is different for each type of conic

e Conic

e=0 Circle

0<e<1 Ellipse

e=1 Parabola

e>1 Hyperbola

Identifying a conic

• All conics can be written in the form • The discriminant of the conic is (looks familiar?)• If the discriminant is positive, then the conic is a hyperbola• If the discriminant is zero, then it is a parabola• If the discriminant is negative, then it is an ellipse• If then it is a circle

Ellipses

• Defined as the locus of points on a plane whose sum of the distances to two points is constant• Focal Length (c) • Eccentricity • Directrix • Latus rectum • Area

Hyperbolas

• Locus of points on a plane whose differences of the distances between two points is constant

• Focal Length (c) • Eccentricity • Directrix • Latus rectum

Parabolas

• Locus of points at an equal distance from a point an a line

• Focal length = p• Latus rectum 4p• All parabolas have a eccentricity of 1

Rotation of axis

• If there is a B term in the standard form of a conic, then there is a rotation. • To find the angle, find θ in this equation:

• Then Substitute in and

Degenerate Conics

• Hyperbolae can degenerate into two intersecting lines, two parallel lines• Parabolae can degenerate to two parallel lines or a double line• Ellipses can degenerate to two parallel lines or the double line• If the determinant of the following matrix is zero, then the conic is

degenerate

Practice problems

• Identify the following non-degenerate conic by using the discriminant:

• A) Hyperbola B) Circle C) Non-circular Ellipse D) Cycloid E) NOTA

Practice Problem 2

• James graphed the following equation on the Argand diagram: . What conic shape did James just graph?

Practice Problem 3

• Find the area and eccentricity of the following conic