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Chapter 10: Problem 79

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

Short Answer

Expert verified
After following the described steps to convert the polar equation to rectangular form, you would get a resulting equation matching the rectangular coordinate system.

Step by step solution

01

Substitute sin(theta) with y/r in the given equation

The sine of the angle \(\theta\) (sin\(\theta\)) in polar coordinates can be written as \(y\) divided by \(r\) (\(y\)/\(r\)) in rectangular coordinates. So, substitute \(y\)/\(r\) for sin\(\theta\) in the equation. The equation becomes \(r = \frac{6}{2 - 3\frac{y}{r}}\).
02

Rewrite equation to eliminate the fraction

Rewrite the equation to remove fractions. Multiplying both sides by \(r\) and \(2 - 3\frac{y}{r}\), the equation becomes \(r^2= 6r(2 - 3\frac{y}{r})\).
03

Replace r^2 with x^2 + y^2

In terms of rectangular coordinates, \(r^2\) equals \(x^2 + y^2\). Replace \(r^2\) in the equation, \(x^2 + y^2 = 6r(2 - 3\frac{y}{r})\).
04

Replace r with sqrt(x^2 + y^2)

In rectangular coordinates, \(r = \sqrt{x^2 + y^2}\). Replace \(r\) in the equation with \(\sqrt{x^2 + y^2}\). The equation becomes \(x^2 + y^2 = 6\sqrt{x^2 + y^2}(2 - \frac{3y}{\sqrt{x^2 + y^2}})\).
05

Solve the equation

Simplify the equation to convert completely to rectangular form. This involves multiplying out the bracket, simplifying fractions where possible and finally, rearranging the equation in terms of y.

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Most popular questions from this chapter

Identify the type of conic represented by the equation. Use a graphing utility to confirm your result. $$r=\frac{9}{3-2 \cos \theta}$$

Convert the rectangular equation to polar form. Assume \(a<0\) $$x^{2}+y^{2}-2 a x=0$$

Convert the rectangular equation to polar form. Assume \(a<0\) $$x^{2}+y^{2}-6 x=0$$

Find a polar equation of the conic with its focus at the pole. $$\begin{array}{cc} \text{Conic} & \text{Eccentricity} & \text{Directrix} \\\ \text{Hyperbola} &e=\frac{3}{2}&x=-1\end{array}$$

Identify the type of conic represented by the polar equation and analyze its graph. Then use a graphing utility to graph the polar equation. $$r=\frac{14}{14+17 \sin \theta}$$
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