The chief aim of all investigations of the external world should be to discover the rational order and harmony which has been imposed on it by God and which He revealed to us in the language of mathematics.

Johannes Kepler

sections:

      10.1, 10.2, 10.3, 10.4, 10.5, and 10.6

We now consider generalized quadratic equations in two-dimensions. These result in a family of curves which can be generated by slicing a cone.

Polar coordinates are useful when equations have certain symmetry—and they are simply cool!

objectives:

• identify the general quadratic equation in two variables
• explain what is meant by a conic section
• define the curve ellipse
• identify standard forms of the equation of an ellipse
• reduce a quadratic equation representing an ellipse to standard form
• calculate the foci and the eccentricity of an ellipse
• graph an ellipse
• relate the ellipse to a circle
• define the terms and use the focus and directrix
• define the curve hyperbola
• identify standard forms of the equation of a hyperbola
• reduce a quadratic equation representing a hyperbola to standard form
• calculate the foci of a hyperbola
• graph a hyperbola
• define the curve parabola
• identify standard forms of the equation of a parabola
• reduce a quadratic equation representing a parabola to standard form
• graph a parabola
• identify conic section that contain an xy term
• discuss the rotation of the axes
• transform a conic section to eliminate the xy term

• identify a parametric equation
• graph a plane curve defined by parametric equations
• eliminate the parameter to derive a relationship between the independent and dependent variables
• discuss the advantages of one form compared to the other
• describe the parametric form of the derivative
• find slope to a curve expressed as a set of parametric equations
• compute the acr length
• calculate the area of a surface of revolution

• plot points in a polar coordinate system
• identify alternative forms of the polar coordinates for the same physical point
• handle negative values of r
• change from cartesian to polar coordinates
• change from polar to cartesion
• graph equations in polar coordinates
• state the symmetry conditions
• express the slope in polar form
• find the area of a region bounded by two polar curves
• compute the acr length
• calculate the area of a surface of revolution
• express the conic sections in polar form
• classify the conic sections by eccentricity
• graph conic sections in polar form
• use Kepler's Laws