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Chapter 10: Problem 80

Convert the polar equation to rectangular form. $$r=\frac{6}{2 \cos \theta-3 \sin \theta}$$

Short Answer

Expert verified
The rectangular form of the given polar equation is \(y = 2x - 3\)

Step by step solution

01

Write down the relations between Polar and Rectangular coordinates

The following relationships exist between polar and rectangular coordinates: \(x = r\cos \theta\) and \(y = r\sin \theta\). These will be used to convert from polar form to rectangular form.
02

Substitute the value of r into the polar to rectangular relations

Use the provided polar equation and substitute \(r=\frac{6}{2\cos \theta-3\sin \theta}\) into both relations from step 1, as follows : \(x = \frac{6\cos \theta}{2\cos \theta-3\sin \theta}\) and \(y = \frac{6\sin \theta}{2\cos \theta-3\sin \theta}\). After simplification you will have: \(x = \frac{6\cos \theta}{2\cos \theta-3\sin \theta}\) and \(y = 2x-3\).
03

Solve for y in the rectangular form

Equating the two relations allows us to solve for y. On equating, we have \(y = 2x-3\).\This is the required rectangular form.

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Most popular questions from this chapter

Use a graphing utility to graph the rotated conic. $$r=\frac{10}{3+9 \sin (\theta+2 \pi / 3)}$$

Describe the graph of the polar equation and find the corresponding rectangular equation. Sketch its graph. $$r=6$$

Find a polar equation of the conic with its focus at the pole. $$\begin{array}{cc} \text{Conic} & \text{Vertex or Vertices} \\\ \text{Ellipse} &(20,0),(4, \pi)\end{array}$$

Determine whether the statement is true or false. Justify your answer. A circle is a degenerate conic.

Identify the type of conic represented by the equation. Use a graphing utility to confirm your result. $$r=\frac{6}{2+\sin \theta}$$
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