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Question 10.5.4: Converting a polar conic to rectangular coordinates Convert ...

Converting a polar conic to rectangular coordinates

Convert r=\frac{4}{2-2 \sin \theta} to rectangular coordinates and identify the conic. Sketch the graph.

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Convert to rectangular coordinates as follows:

\begin{aligned} r &=\frac{4}{2-2 \sin \theta} \\ 2 r-2 r \sin \theta &=4 \\ r &=2+r \sin \theta \\ \sqrt{x^{2}+y^{2}} &=2+y \\ x^{2}+y^{2} &=4+4 y+y^{2} \\ x^{2}-4 y-4 &=0 \\ y &=\frac{1}{4} x^{2}-1 \end{aligned}

The last equation is the equation for a parabola opening upward, as shown in Fig. 10.53.

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