M408M Learning Module Pages
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Chapter 10: Parametric Equations
and Polar Coordinates
Learning module LM 10.1:
Parametrized Curves:
Learning module LM 10.2: Calculus with Parametrized Curves:
Learning module LM 10.3: Polar Coordinates:
Learning module LM 10.4: Areas and Lengths of Polar Curves:
Learning module LM 10.5: Conic Sections:
Learning module LM 10.6: Conic Sections in Polar Coordinates:
Foci and directrices
Visualizing eccentricity
Polar equations for conic sections
Astronomy
Chapter 12: Vectors and the Geometry of Space
Chapter 13: Vector Functions
Chapter 14: Partial Derivatives
Chapter 15: Multiple Integrals


Polar equations for conic sections
Polar Equations for Conic Sections
We will work with conic sections with a focus at the origin.
Polar equations of conic sections: If the
directrix is a distance $d$ away, then the polar form of a conic
section with eccentricity $e$ is
$$r(\theta) = \frac{ed}{1e
\cos(\theta\theta_0)},$$
where the constant $\theta_0$ depends on the
direction of the directrix.

This formula applies to all conic sections. The only difference
between the equation of an ellipse and the equation of a parabola and
the equation of a hyperbola is the value of the eccentricity $e$.
There are four important special cases:
 If the directrix is the line $x=d$, then we have $\displaystyle{r
= \frac{ed}{1+e\cos(\theta)}.}$
 If the directrix is the line $x=d$, then we have $\displaystyle{r =
\frac{ed}{1e\cos(\theta)}.}$
 If the directrix is the line $y=d$, then we have $\displaystyle{r =
\frac{ed}{1+e\sin(\theta)}.}$
 If the directrix is the line $y=d$, then we have $\displaystyle{r =
\frac{ed}{1e\sin(\theta)}.}$

In all cases, the curve opens out in the direction opposite to the directrix.
