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Chapter 8: Problem 33

Convert the polar equation to rectangular form and sketch its graph. $$ r=3 \sec \theta $$

Short Answer

Expert verified
The rectangular form of \( r = 3 \ \sec \theta \) is \( x=3 \). The graph of this equation is a vertical line at \( x=3 \).

Step by step solution

01

Conversion from polar to rectangular form using identities

To convert this polar form to rectangular form, substitute \( r \cos \theta \) for x and \( r \sin \theta \) for y. But before that, implement the reciprocal identity for the secant function, \( \sec \theta = \frac{1}{\cos \theta} \). Therefore, the equation becomes \( r = \frac{3}{\cos \theta} \). Now, simply substitute \( r \cos \theta \) for x in the equation, yielding \( r \cos \theta = 3 \). Implementing the fact that \( x = r \cos \theta \), the rectangular form equation is \( x=3 \).
02

Sketching the rectangular form

The graph of \( x=3 \) is a vertical line that crosses the x-axis at \( x=3 \). This vertical line extends indefinitely in the positive and negative y direction.

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Most popular questions from this chapter

Give the integral formulas for the area of the surface of revolution formed when the graph of \(r=f(\theta)\) is revolved about (a) the \(x\) -axis and (b) the \(y\) -axis.

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. \(r=\frac{6}{1+\cos \theta}\)

Eliminate the parameter and obtain the standard form of the rectangular equation. $$ \text { Circle: } x=h+r \cos \theta, \quad y=k+r \sin \theta $$

In Exercises \(17-20,\) use a graphing utility to graph the polar equation. Identify the graph. \(r=\frac{-3}{2+4 \sin \theta}\)

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.) \(\frac{\text { Conic }}{\text { Hyperbola }} \quad \frac{\text { Eccentricity }}{e=\frac{3}{2}} \quad \frac{\text { Directrix }}{x=-1}\)
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