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Chapter 8: Problem 24

Convert the rectangular equation to polar form and sketch its graph. $$ x y=4 $$

Short Answer

Expert verified
The polar form of given rectangular equation is \(r^2 = \frac{8}{\sin(2\theta)}\) if \(\sin(2\theta) 鈮� 0\). The graph is a lemniscate, which forms loops in the positive and negative regions.

Step by step solution

01

Converting rectangular equation to polar form

Replace \(x\) and \(y\) in the given equation \(xy = 4\) by corresponding polar forms. So, \(r\cos(\theta) \cdot r\sin(\theta) = 4\). This leads to \(r^2 \sin(\theta) \cos(\theta) = 4\).
02

Simplifying the polar equation

The double-angle identity for \(\sin(2\theta) = 2 \sin(\theta) \cos(\theta)\) can be used to simplify the equation. With this identity, the equation then becomes \(r^2 \sin(2\theta) = 8\). This can be further simplified to \(r^2 = \frac{8}{\sin(2\theta)}\) if \(\sin(2\theta) 鈮� 0\).
03

Sketching the Polar Graph

This graph represents a lemniscate when \(r 鈮� 0\) and \(\sin(2\theta) 鈮� 0\). The graph passes the origin at \(\theta = 45掳\) and \(\theta = 135掳\). The loop in the positive region is contained within \(0 < \theta < 90掳\), and the loop in the negative region is contained within \(90掳 < \theta < 180掳\).

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Most popular questions from this chapter

In Exercises 43-46, find the area of the surface formed by revolving the curve about the given line. $$ \begin{array}{lll} \underline{\text { Polar Equation }} & \underline{\text { Interval }} & \underline{\text { Axis of Revolution }} \\ r=a \cos \theta & 0 \leq \theta \leq \frac{\pi}{2} & \theta=\frac{\pi}{2} \end{array} $$

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. $$ \begin{array}{l} x=4+2 \cos \theta \\ y=-1+\sin \theta \end{array} $$

Eliminate the parameter and obtain the standard form of the rectangular equation. $$ \begin{aligned} &\text { Line through }\left(x_{1}, y_{1}\right) \text { and }\left(x_{2}, y_{2}\right)\\\ &x=x_{1}+t\left(x_{2}-x_{1}\right), \quad y=y_{1}+t\left(y_{2}-y_{1}\right) \end{aligned} $$

Sketch the curve represented by the parametric equations (indicate the orientation of the curve), and write the corresponding rectangular equation by eliminating the parameter. $$ x=e^{t}, \quad y=e^{3 t}+1 $$

In Exercises \(7-16,\) find the eccentricity and the distance from the pole to the directrix of the conic. Then sketch and identify the graph. Use a graphing utility to confirm your results. \(r(3-2 \cos \theta)=6\)
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