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Chapter 6: Problem 29

Test for symmetry and then graph each polar equation. $$r^{2}=9 \cos 2 \theta$$

Short Answer

Expert verified
The polar equation \(r^{2}=9 \cos 2 \theta\) is symmetrical with respect to the origin and its graph is an ellipse centred at the origin with its major axis on the y-axis and minor axis on the x-axis.

Step by step solution

01

Testing for Symmetry

There are three types of symmetry for polar equations: symmetry about the x-axis, the y-axis, and the origin. For this equation, since it contains cos(2胃), it does have symmetry. Precisely, it is symmetric about the origin, as replacing \(r\) and \(\theta\) with \(-r\) and \(\theta+蟺\) respectively yields the same equation.
02

Converting from Polar to Cartesian Coordinates

It might be helpful to convert this equation to Cartesian coordinates for graphing purposes. The conversion formulas are \(r^{2}=x^{2}+y^{2}\) and \(cos 2 \theta =(1-2y^{2})\). Replacing these into our polar equation, we get \(x^{2}+y^{2}=9(1-2y^{2})\). It simplifies to \(x^{2}+y^{2}=9-18y^{2}\), which further simplifies to \(x^{2}+18y^{2}-9=0\). This is an equation of an ellipse in standard form.
03

Graphing the Equation

The graph depicts an ellipse with its centre at the origin (0,0). The major axis lies along the y-axis and has a length of \(\sqrt{2}\) in each direction from the centre, that's why the range of y is from -1/3 to 1/3. The minor axis lies along the x-axis and spans from -3 to 3.

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Most popular questions from this chapter

A force is given by the vector \(\mathbf{F}=3 \mathbf{i}+2 \mathbf{j} .\) The force moves an object along a straight line from the point (4,9) to the point \((10,20) .\) Find the work done if the distance is measured in feet and the force is measured in pounds.

Graph the spiral \(r=\theta .\) Use a [-48,48,6] by [-30,30,6] viewing rectangle. Let \(\theta \min =0\) and \(\theta \max =2 \pi,\) then \(\theta \min =0\) and \(\theta \max =4 \pi,\) and finally \(\theta \min =0\) and \(\theta \max =8 \pi\)

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A force of 80 pounds on a rope is used to pull a box up a ramp inclined at \(10^{\circ}\) from the horizontal. The rope forms an angle of \(33^{\circ}\) with the horizontal. How much work is done pulling the box 25 feet along the ramp?

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I'm working with a polar equation that failed the symmetry test with respect to \(\theta=\frac{\pi}{2},\) so my graph will not have this kind of symmetry.
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