/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 69 Describe the graph of the polar ... [FREE SOLUTION] | 91影视

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Chapter 6: Problem 69

Describe the graph of the polar equation and find the corresponding rectangular equation. Sketch its graph. $$r=3 \sec \theta$$

Short Answer

Expert verified
The graph of the polar equation \( r=3 \sec \theta \) in rectangular form is a vertical line intersecting the x-axis at \( x=3 \).

Step by step solution

01

Convert Polar Equation to Rectangular Form

Start by noting the formula to convert polar coordinates (r, 胃) to rectangular coordinates (x, y): \( r = \sqrt{x^2 + y^2} \) and \( x = r \cos \theta \). Since \( r = 3 \sec \theta \), we can replace \( \sec \theta \) with \( 1/ \cos \theta \). This gives \( r = 3 / \cos \theta \). Multiply each side by \( \cos \theta \), giving \( r \cos \theta = 3 \). Since \( x = r \cos \theta \), our rectangular coordinate equivalent is simply \( x=3 \).
02

Describe the Rectangular Graph

The rectangular equation \( x=3 \) represents a vertical line that intersects the x-axis at x=3. This line stretches infinitely in the positive and negative y-direction.
03

Sketch the Graph

Plot a vertical line at \( x=3 \) on a graph, with the line extending indefinitely in both the positive and negative y-directions. This line graphically represents the equation \( x=3 \).

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Most popular questions from this chapter

In Exercises 33-48, find a polar equation of the conic with its focus at the pole. $$ \begin{array}{ll} {\text { Conic }} & \text { Vertex or Vertices } \\ \text { Ellipse } & (2,0),(10, \pi) \\ \end{array} $$

In Exercises 17-40, sketch the graph of the polar equation using symmetry, zeros, maximum \(r\)-values, and any other additional points. \(r=5\)

In Exercises 33-48, find a polar equation of the conic with its focus at the pole. $$ \begin{array}{ll} {\text { Conic }} & \text { Vertex or Vertices } \\ \text { Parabola } & (5, \pi) \\ \end{array} $$

In Exercises 7-12, test for symmetry with respect to \(\theta=\pi / 2\), the polar axis, and the pole. \(r=5+4 \cos \theta\)

\(r=3 \cos 2 \theta\)
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