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Chapter 6: Problem 64

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 equation is \(2x - 3y = \sqrt{(x^2+y^2)}\)

Step by step solution

01

Isolate the denominator

In order to move forward with substituting \(r\), we first isolate the denominator :- \[ 2 \cos \theta - 3 \sin \theta = r/3 \]
02

Substitute the polar-rectangular relationships

We replace \(r\), \(\cos\), and \(\sin\) with their rectangular equivalents: \[ 2 \frac{x}{\sqrt{(x^2+y^2)}} - 3 \frac{y}{\sqrt{(x^2+y^2)}} = \frac{\sqrt{(x^2+y^2)}}{3} \]
03

Simplify the equation

To convert fully into rectangular form we clear the denominator by multiplying the entire equation by \(\sqrt{(x^2+y^2)}\). This yields the rectangular form: \[ 2x - 3y = \sqrt{(x^2+y^2)}\]

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Most popular questions from this chapter

\(r=3(2-\sin \theta)\)

In Exercises 83 and 84 , find the exact values of \(\sin 2 u\), \(\cos 2 u\), and \(\tan 2 u\) using the double-angle formulas. $$ \tan u=-\sqrt{3}, \frac{3 \pi}{2}

In Exercises 25-28, use a graphing utility to graph the polar equation. Identify the graph. $$ r=\frac{4}{1-2 \cos \theta} $$

The equation $$ r=\frac{e p}{1 \pm e \sin \theta} $$ is the equation of an ellipse with \(e<1\). What happens to the lengths of both the major axis and the minor axis when the value of \(e\) remains fixed and the value of \(p\) changes? Use an example to explain your reasoning.

\(r=1-2 \cos \theta\)
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