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Chapter 10: Problem 18

Test for symmetry with respect to the line \(\theta=\pi / 2,\) the polar axis, and the pole. $$r=\frac{2}{1+\sin \theta}$$

Short Answer

Expert verified
The given polar equation \(r=\frac{2}{1+\sin \theta}\) is symmetric with respect to the polar axis, but not with respect to the line \(\theta=\pi / 2\) or the pole.

Step by step solution

01

Symmetry with respect to the line \(\theta=\pi / 2\)

To check for symmetry with respect to the line \(\theta=\pi / 2\), one must replace \(\theta\) by \(\pi -\theta\) in the original equation. If the resulting equation is equivalent to the original one, then the given equation is symmetric with respect to the line \(\theta=\pi / 2\).
02

Symmetry with respect to the polar axis

To check for symmetry with respect to the polar axis, \(\theta\) should be replaced with \(-\theta\) in the original equation. If the resulting equation is the same as the initial equation, it means that the original polar function has symmetry with respect to the polar axis.
03

Symmetry with respect to the pole

In order to see if the equation is symmetric with respect to the pole, \(r\) must be replaced with \(-r\) in the original equation. If the equation obtained is equivalent to the original one, then the polar equation is symmetric relative to the pole. However, in this case, since it's impossible to substitute \(r\) with \(-r\) without changing the equation, it can be concluded that it doesn't have symmetry with respect to the pole.

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Most popular questions from this chapter

Check for symmetry with respect to both axes and to the origin. Then determine whether the function is even, odd, or neither. $$f(x)=\frac{4 x^{2}}{x^{2}+1}$$

Check for symmetry with respect to both axes and to the origin. Then determine whether the function is even, odd, or neither. $$(x-2)^{2}=y+4$$

The graph of \(r=f(\theta)\) is rotated about the pole through an angle \(\phi .\) Show that the equation of the rotated graph is \(r=f(\theta-\phi)\).

Check for symmetry with respect to both axes and to the origin. Then determine whether the function is even, odd, or neither. $$y=e^{x}$$

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