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Chapter 6: Problem 65

Describe the graph of the polar equation and find the corresponding rectangular equation. Sketch its graph. $$r=6$$

Short Answer

Expert verified
The graph of the polar equation \(r = 6\) is a circle with a radius of 6 centered at the origin. Its corresponding rectangular equation is \(x^{2} + y^{2} = 36\).

Step by step solution

01

Understand the Polar Equation

The equation \(r = 6\) is a polar equation. In polar coordinates, r represents the radial distance from the origin. Thus, this equation describes a circle with a radius of 6, centered at the origin.
02

Convert to Rectangular Coordinates

To convert this polar equation to a rectangular one, we can use the standard conversion equation \(r^{2} = x^{2} + y^{2}\). Here, since \(r = 6\), if we square both sides of the equation, we get \(36 = x^{2} + y^{2}\). This is the corresponding rectangular equation.
03

Sketching the Graph

This equation in rectangular coordinates can be plotted as a circle with radius 6 and the center at the origin (0,0). A perfect circle can be formed by drawing points at every angle from the origin with a distance of 6 units, as defined by radius.

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Most popular questions from this chapter

In Exercises 79-82, find the exact value of the trigonometric function given that \(u\) and \(v\) are in Quadrant IV and \(\sin u=-\frac{3}{5}\) and \(\cos v=1 / \sqrt{2}\). $$ \cos (u+v) $$

\(r=3-4 \cos \theta\)

\(r^{2}=16 \cos 2 \theta\)

\(r=4+3 \cos \theta\)

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