15.4 Outline
- Introduction
- definition
- model
- Geometric definition of conic sections
- parabola
- focus
- directrix
- axis
- vertex
- ellipse
- foci
- major axis
- center
- minor axis
- circle
- center
- radius
- hyperbola
- transverse axis
- center
- conjugate axis
- parabola
- Algebraic definition of conic sections
- general form
- first-degree equation
- second-degree equation
- line
- parabola
- ellipse
- circle
- hyperbola
- general form
- Graphing conic sections
- standard form
- ellipses
- standard-form equations
- equation of a circle
- horizontal ellipse
- vertical ellipse
- eccentricity
- aphelion
- perihelion
- applications
- hyperbolas
- standard-form equations
- vertices
- horizontal hyperbola
- vertical hyperbola
- length of axis
- applications
- Parabolic reflectors
15.4 Essential Ideas
Geometric definition of the conic sections:
A parabola is the set of all points in the plane equidistant from a given point (called the focus) and a given line (called the directrix).
An ellipse is the set of all points in a plane such that, for each point on the ellipse, the sum of its distances from two fixed points (called the foci) is a constant.
A circle (a special type of an ellipse) is the set of all points in a plane a given distance from a given point.
A hyperbola is the set of all points in a plane such that, for each point on the hyperbola, the difference of its distances from two fixed points (the foci) is a constant.
Algebraic definition of the conic sections:
The general form of the equation of a conic section is
Ax2 + Bxy + Cy2 + Dx + Ey + F = 0
where A, B, C, D, E, and F are real numbers and (x, y) is any point on the curve.
This equation is called a: first-degree equation if A = B = C =0
second-degree equation otherwise.
If B = 0, then we classify the conic section as follows: A = C = 0,
then the conic section is a line;
A = 0 and C is not equal to 0 or if A is not equal to 0 and C = 0, it is a parabola; A and C have the same sign, it is an ellipse;
A = C is a circle; and A and C have opposite signs, it is a hyperbola.
If B is not equal to 0, then the conic is rotated.