Example 4 from Algebra and Trigonometry, 12.5 Conic Sections in Polar Coordinates

Graphing an Ellipse in Polar Form

Graph $r = \frac{10}{5 - 4 \cos} .$

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Solution
First, we rewrite the conic in standard form by multiplying the numerator and denominator by the reciprocal of 5, which is $\frac{1}{5} .$

$\begin{array}{l} \begin{array}{l} r = \frac{10}{5 - 4 \cos θ} = \frac{10 (\frac{1}{5})}{5 (\frac{1}{5}) - 4 (\frac{1}{5}) \cos θ} \\ r = \frac{2}{1 - \frac{4}{5} \cos} \end{array} \end{array}$

Because $e = \frac{4}{5}, e < 1,$ so we will graph an ellipse with a focus at the origin. The function has a $cos θ,$ and there is a subtraction sign in the denominator, so the directrix is $x = - p .$

$\begin{array}{l} 2 = e p \\ 2 = (\frac{4}{5}) p \\ 2 (\frac{5}{4}) = p \\ \frac{5}{2} = p \end{array}$
The directrix is $x = - \frac{5}{2} .$

Plotting a few key points as in Table 3 will enable us to see the vertices. See Figure 6.

A B C D

$θ$ $0$ $\frac{π}{2}$ $π$ $\frac{3 π}{2}$

$r = \frac{10}{5 - 4 \cos}$ $10$ $2$ $\frac{10}{9} \approx 1.1$ $2$

Table 3

Figure 6

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

Graph $r = \frac{2}{4 - \cos θ} .$

How did it go?