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Chapter 6: Problem 66

Describe the graph of the polar equation and find the corresponding rectangular equation. Sketch its graph. $$r=8$$

Short Answer

Expert verified
The graph of the polar equation \(r=8\) is a circle centered at the origin with a radius of 8 units. The equivalent rectangular equation is \(x^2 + y^2 = 64\).

Step by step solution

01

Polar Equation Analysis

In polar coordinates, \(r\) represents the distance from the origin to a point, while \(\theta\) represents the angle made by the point with the positive x-axis. Since it's given that \(r=8\), this represents a set of all points which are at a distance of 8 units from the origin, which is a clear description of a circle with a radius of 8 units centered at the origin (0,0).
02

Conversion to Rectangular Coordinates

To convert the polar equation \(r=8\) to rectangular coordinates, we can use the relationship between polar and rectangular coordinates: \(r^2 = x^2 + y^2\). Replacing \(r\) with 8, the equation becomes \(8^2 = x^2 + y^2\), or \(x^2 + y^2 = 64\). This is the equation of a circle with radius 8 in rectangular coordinates
03

Sketching the graph

The graph is a circle centered at the origin (0,0) with a radius of 8 units. To sketch the graph, plot the center at (0,0), then sketch the circle by plotting points 8 units away from the origin in all directions.

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