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Chapter 8: Problem 67

Sketch a graph of the polar equation. $$ r=5 $$

Short Answer

Expert verified
The graph of the polar equation \( r = 5 \) is a circle with a radius of 5 units, centered at the origin, on the polar axis.

Step by step solution

01

Understand the basics of the polar coordinate system

A polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. Here, the reference point is the origin (0,0), and the reference direction is the positive x-axis. The polar coordinates (r,胃) are then given by the radius r and the angle 胃.
02

Translate the polar equation to corresponding graph representation

The given equation is \( r = 5 \), which describes a circle of radius 5 centered at the origin. There is no 胃 term in the equation, which means that the circle is not rotated - it lies in the standard position, centered at the origin.
03

Sketch the graph

Draw the polar axis, which is a horizontal line called the initial line. Choose a length for 5 units on the paper and mark it on the initial line. Draw a circle with radius 5 units centered at the origin. This circle represents the graph of the polar equation \( r = 5 \).

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Most popular questions from this chapter

Write the equation for the ellipse rotated \(\pi / 4\) radian clockwise from the ellipse \(r=\frac{5}{5+3 \cos \theta}\).

Sketch the curve represented by the parametric equations (indicate the orientation of the curve), and write the corresponding rectangular equation by eliminating the parameter. $$ x=|t-1|, \quad y=t+2 $$

Use a graphing utility to graph the curve represented by the parametric equations. Indicate the direction of the curve. Identify any points at which the curve is not smooth. $$ \text { Hypocycloid: } x=3 \cos ^{3} \theta, \quad y=3 \sin ^{3} \theta $$

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. \(r=\frac{-1}{1-\sin \theta}\)

Sketch the curve represented by the parametric equations (indicate the orientation of the curve), and write the corresponding rectangular equation by eliminating the parameter. $$ x=t^{3}, \quad y=\frac{t^{2}}{2} $$
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