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Chapter 8: Problem 21

Convert the rectangular equation to polar form and sketch its graph. $$ y=4 $$

Short Answer

Expert verified
The polar form of the rectangular equation \(y = 4\) is \(r\sin(\theta) = 4\). The graph will be a circle with the centre at the origin and radius 4.

Step by step solution

01

Identify the equation in rectangular form

The given equation is \(y = 4\). This is clearly a horizontal line where y is always 4, regardless of the value of x.
02

Convert the rectangular equation to polar form

For the polar coordinates, \((r, \theta)\), \(y = r\sin(\theta)\). Hence, for the given equation, we substitute y with r*sin(theta) to get \(r\sin(\theta) = 4\).
03

Sketch its graph

The polar form of the equation, \(r\sin(\theta) = 4\), represents a line in polar coordinates. This line is the set of all points which are exactly 4 units from the origin with an angle \(\theta\) of the polar coordinates ranging all possible angles from 0 to \(2\pi\). Therefore, the graph will be a circle with the centre at the origin and radius 4.
04

Final confirmation

Confirm that the polar equation and the graph correspond with the original rectangular equation \(y = 4\). It's clear that they do, because the polar equation simplifies to the rectangular equation if we use the relationship \(y = r\sin(\theta)\), and the graph shows a horizontal line 4 units above the origin.

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Most popular questions from this chapter

In Exercises \(27-38,\) find a polar equation for the conic with its focus at the pole. (For convenience, the equation for the directrix is given in rectangular form.) \(\frac{\text { Conic }}{\text { Hyperbola }} \quad \frac{\text { Eccentricity }}{e=\frac{3}{2}} \quad \frac{\text { Directrix }}{x=-1}\)

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. \(r=\frac{5}{5+3 \sin \theta}\)

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. \(r(3-2 \cos \theta)=6\)

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

Sketch the curve represented by the parametric equations (indicate the orientation of the curve), and write the corresponding rectangular equation by eliminating the parameter. $$ x=\sec \theta, \quad y=\cos \theta $$
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