Converting a Conic in Polar Form to Rectangular Form

Convert the conic $r = \frac{1}{5 - 5 \sin θ}$ to rectangular form.

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
We will rearrange the formula to use the identities $r = \sqrt{x^{2} + y^{2}}, x = r \cos θ, and y = r \sin θ .$

$\begin{array}{l} r = \frac{1}{5 - 5 \sin θ} \\ r \cdot (5 - 5 \sin θ) = \frac{1}{5 - 5 \sin θ} \cdot (5 - 5 \sin θ) & Eliminate the fraction . \\ 5 r - 5 r \sin θ = 1 & Distribute . \\ 5 r = 1 + 5 r \sin θ & Isolate 5 r . \\ 25 r^{2} = {(1 + 5 r \sin θ)}^{2} & Square both sides . \\ 25 (x^{2} + y^{2}) = {(1 + 5 y)}^{2} & Substitute r = \sqrt{x^{2} + y^{2}} and y = r \sin θ . \\ 25 x^{2} + 25 y^{2} = 1 + 10 y + 25 y^{2} & Distribute and use FOIL . \\ 25 x^{2} - 10 y = 1 & Rearrange terms and set equal to 1 . \end{array}$

Your turn

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

Convert the conic $r = \frac{2}{1 + 2 \cos}$ to rectangular form.

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