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Chapter 8: Problem 23

Convert the rectangular equation to polar form and sketch its graph. $$ 3 x-y+2=0 $$

Short Answer

Expert verified
The polar form of the given rectangle equation is \(r = -2/(3\cos\theta - \sin\theta)\). The sketch involves a curve, with its maximum in the \(3\pi/4\) direction, extends indefinitely in the \(\pi/4\) and \(-3\pi/4\) directions. Where points are negative, reflect them across the origin.

Step by step solution

01

Convert Rectangular Equations into Polar form

In the given equation \(3x - y + 2 = 0\), replace \(x\) with \(r \cos \theta\) and \(y\) with \(r \sin \theta\). The equation becomes \(3r \cos \theta - r \sin \theta + 2 = 0\). To simplify further, factor out the \(r\) which yields \(r(3 \cos \theta - \sin \theta )+2=0\).
02

Solve for r

For the graph, it is easier to have \(r\) on one side of the equation. Subtract 2 from both side to get \(r(3 \cos \theta - \sin\theta) = -2\). Then divide both sides by \((3 \cos \theta - \sin \theta)\) to isolate \(r\), producing \(r = -2/(3 \cos \theta - \sin \theta)\).
03

Sketch the Graph

The graph sketching involves plotting the points of \(r\) and \(\theta\) on a polar grid. As \(\theta\) varies from \(0\) to \(360^{\circ}\) or \(0\) to \(2\pi\), mark the corresponding \(r\) values. Remember, \(r<0\) means the point is in the opposite direction. Properly interpret points where \(r\) is negative by reflecting over the origin and acquire smooth curve by joining these points.

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Most popular questions from this chapter

Find the area of the circle given by \(r=\sin \theta+\cos \theta\). Check your result by converting the polar equation to rectangular form, then using the formula for the area of a circle.

In Exercises \(27-38,\) find a polar equation for the conic with its focus at the pole. (For convenience, the equation for the directrix is given in rectangular form.) \(\frac{\text { Conic }}{\text { Ellipse }} \quad \frac{\text { Eccentricity }}{e=\frac{3}{4}} \quad \frac{\text { Directrix }}{x=-2}\)

Sketch the curve represented by the parametric equations (indicate the orientation of the curve), and write the corresponding rectangular equation by eliminating the parameter. $$ x=\tan ^{2} \theta, \quad y=\sec ^{2} \theta $$

Sketch the curve represented by the parametric equations (indicate the orientation of the curve), and write the corresponding rectangular equation by eliminating the parameter. $$ x=2 t^{2}, \quad y=t^{4}+1 $$

Use a graphing utility to graph the curve represented by the parametric equations (indicate the orientation of the curve). Eliminate the parameter and write the corresponding rectangular equation. $$ x=e^{2 t}, \quad y=e^{t} $$
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