Calc 2 Lecture NotesSection 9.7Page 1 of 11

Section 9.7: Conic Sections in Polar Coordinates

Big idea: The ellipse, hyperbola, and parabola all have the same equation in polar coordinates that is parameterized by a single constant called the eccentricity. This is significant for physics, since one can calculate the eccentricity for any object moving under the influence of a “central force,” which means that the trajectory of that object can be predicted easily.

Big skill:. You should be able to plot the conic sections given in polar form, and convert the rectangular forms of their equations to polar and parametric forms.

Theorem 7.1: Eccentricity of the conic sections.

The set of all points whose distance to the focus is the product of the eccentricity e and the distance to a directrix is:

  • An ellipse for 0 < e < 1.
  • A parabola if e = 1.
  • A hyperbola if e > 1.

A circle has eccentricity 0 (because a = b…).

Practice:Show that theorem 7.1 is true, and then convert the Cartesian equation to polar form. Assume that the focus is at the origin, and the directrix is at x = d > 0 (Note: shifting the origin of the coordinate system to the focus is a common practice for many central force problems, because this is the location of the central force; i.e., the sun or something like that).

For e = 1 (parabola):

For 0 < e < 1 (ellipse):

 1 – e2 > 0…

For e 1 (hyperbola):

 1 – e2 < 0…


Polar form of the conic sections in this orientation:

Practice:

Find the polar equations for the conic sections with focus at (0, 0), directrix x = 2, and eccentricities of e = 0.4, e = 0.8, e = 1, e = 1.2, e = 2. Then graph the equations.

For a parabola, the eccentricity is .

The rectangular coordinate equation for this orientation is

For an ellipse, the range of the eccentricity is 0 < e < 1.

The closer e is to zero, the closer the ellipse is to a circle. The closer e is to 1, the more the ellipse stretches out.

The rectangular coordinate equation for this orientation is , where , which implies that and the directrix is at

For a hyperbola, the eccentricity has value e > 1.

The rectangular coordinate equation for this orientation is , where ,

which implies that , and that the directrix is at

Theorem 7.2: Polar equations for conic sections with different directrixes.

The conic section with eccentricity e > 0, focus (0, 0) and the indicated directrix has the polar equation:

  • , if the directrix is the line x = d > 0.
  • , if the directrix is the line x = d < 0.
  • , if the directrix is the line y = d > 0.
  • , if the directrix is the line y = d 0.

Practice: Graph and interpret the following conic sections:

; ;

Comparison of representations for the conic sections:

Parabola

Rectangular Representation:
Polar Representation:
Parametric Representation: / Circle

Rectangular Representation:
Polar Representation:
Parametric Representation:
for
Or for
Ellipse

Rectangular Representation:
Polar Representation:
for 0 < e < 1
Parametric Representation:
for
Or for / Hyperbola

Rectangular Representation:
Polar Representation:
for e > 1
Parametric Representation:
for (right branch only)
Or for
Or for

Show that rotating the graph of the unit hyperbola by 45 results in the graph of the reciprocal function .