Ch.12 - Parametric and Polar Curves

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch.12 - Parametric and Polar Curves

Problem 12.2.5

Chapter 12, Problem 12.2.5

What is the polar equation of the vertical line x = 5?

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Recall the relationship between Cartesian coordinates \((x, y)\) and polar coordinates \((r, \theta)\): \(x = r \cos{\theta}\) and \(y = r \sin{\theta}\).

Given the vertical line \(x = 5\), substitute \(x\) with \(r \cos{\theta}\) to rewrite the equation in polar form: \(r \cos{\theta} = 5\).

To express \(r\) explicitly, solve for \(r\): \(r = \frac{5}{\cos{\theta}}\).

Recognize that \(\frac{1}{\cos{\theta}}\) is \(\sec{\theta}\), so the polar equation can also be written as \(r = 5 \sec{\theta}\).

Note that this equation represents all points where the radius \(r\) and angle \(\theta\) satisfy the condition for the vertical line \(x=5\) in polar coordinates.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Polar Coordinates

Polar coordinates represent points in the plane using a radius and an angle, denoted as (r, θ), where r is the distance from the origin and θ is the angle measured from the positive x-axis. This system is useful for describing curves and lines in terms of angles and distances rather than x and y coordinates.

Conversion Between Cartesian and Polar Coordinates

To convert between Cartesian (x, y) and polar (r, θ) coordinates, use the formulas x = r cos(θ) and y = r sin(θ). These relationships allow us to rewrite equations given in x and y into polar form by substituting x and y with their polar equivalents.

Equation of a Vertical Line in Polar Form

A vertical line x = a in Cartesian coordinates can be expressed in polar form by substituting x = r cos(θ). Thus, the equation becomes r cos(θ) = a, which relates r and θ for points on the vertical line. This form helps analyze and graph vertical lines using polar coordinates.

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45–60. Areas of regions Find the area of the following regions.

The region outside the circle r = 1/2 and inside the circle r = cos θ

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Explain why the slope of the line θ=π/2 is undefined.

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Use calculus to find the arc length of the line segment x=3t+1, y=4t, for 0≤t≤1. Check your work by finding the distance between the endpoints of the line segment.

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80–83. Equations of circles Use the results of Exercises 78–79 to describe and graph the following circles.

r² - 8r cos(θ - π/2) = 9

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31–36. Eliminating the parameter Eliminate the parameter to express the following parametric equations as a single equation in x and y.

x=tan t, y=sec ² t−1

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25–28. Horizontal and vertical tangents Find the points at which the following polar curves have horizontal or vertical tangent lines.

r = 4 cos θ

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