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 12.2.6

Chapter 12, Problem 12.2.6

What is the polar equation of the horizontal line y = 5?

Verified step by step guidance

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

Since the line is given by \(y = 5\), substitute \(y\) with \(r \sin{\theta}\) to get the equation \(r \sin{\theta} = 5\).

To express \(r\) in terms of \(\theta\), solve the equation for \(r\): \(r = \frac{5}{\sin{\theta}}\).

Note that this equation is valid for values of \(\theta\) where \(\sin{\theta} \neq 0\), which corresponds to angles where the line \(y=5\) exists in polar coordinates.

Thus, the polar equation of the horizontal line \(y=5\) is \(r = \frac{5}{\sin{\theta}}\).

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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 that are difficult to express in Cartesian 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. Conversely, r = √(x² + y²) and θ = arctan(y/x). These relationships allow expressing Cartesian equations in terms of r and θ.

Equation of a Horizontal Line in Polar Form

A horizontal line y = c can be expressed in polar coordinates by substituting y = r sin θ. Thus, the equation becomes r sin θ = c. This form relates the radius and angle for all points on the horizontal line.

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Tangent line at the origin Find the polar equation of the line tangent to the polar curve r=4cosθ at the origin. Explain why the slope of this line is undefined.

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39–50. Equations of ellipses and hyperbolas Find an equation of the following ellipses and hyperbolas, assuming the center is at the origin.

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15–30. Working with parametric equations Consider the following parametric equations.

a. Eliminate the parameter to obtain an equation in x and y.

b. Describe the curve and indicate the positive orientation.

x = r − 1, y = r³; −4 ≤ r ≤ 4

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Tangent lines for a hyperbola Find an equation of the line tangent to the hyperbola x²/a² + y²/b² = 1 at the point (x₀, y₀)

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33–40. Areas of regions Make a sketch of the region and its bounding curves. Find the area of the region.

The region inside the curve r = √(cos θ)

Textbook Question

15–30. Working with parametric equations Consider the following parametric equations.

a. Eliminate the parameter to obtain an equation in x and y.

b. Describe the curve and indicate the positive orientation.

x = 8 + 2t, y = 1; −∞ < t < ∞