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 12.4.50

Chapter 12, Problem 12.4.50

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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Identify the type of conic section: Since the graph shows a hyperbola centered at the origin with vertices and foci along the y-axis, it is a vertical hyperbola.

Recall the standard form of the equation for a hyperbola centered at the origin with a vertical transverse axis: \(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\).

Determine the value of \(a\): The vertices are at \((0, 6)\) and \((0, -6)\), so the distance from the center to each vertex is \(a = 6\), which means \(a^2 = 36\).

Determine the value of \(c\): The foci are at \((0, 10)\) and \((0, -10)\), so the distance from the center to each focus is \(c = 10\), which means \(c^2 = 100\).

Use the relationship between \(a\), \(b\), and \(c\) for hyperbolas: \(c^2 = a^2 + b^2\). Substitute the known values to solve for \(b^2\): \(100 = 36 + b^2\), then \(b^2 = 100 - 36\).

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Key Concepts

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

Standard Equation of a Hyperbola Centered at the Origin

A hyperbola centered at the origin with a vertical transverse axis has the equation \( \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \). Here, \(a\) is the distance from the center to each vertex along the y-axis, and \(b\) relates to the distance along the conjugate axis. This form is essential to write the equation based on given vertices and foci.

Relationship Between Vertices, Foci, and Parameters \(a\), \(b\), and \(c\)

In a hyperbola, \(a\) is the distance from the center to each vertex, and \(c\) is the distance from the center to each focus. These satisfy the equation \( c^2 = a^2 + b^2 \). Knowing \(a\) and \(c\) allows calculation of \(b\), which is necessary to complete the hyperbola's equation.

Graph Interpretation and Coordinate Identification

Analyzing the graph helps identify key points such as vertices and foci coordinates. For this hyperbola, vertices at (0,6) and (0,-6) give \(a=6\), and foci at (0,10) and (0,-10) give \(c=10\). These values are critical inputs for forming the hyperbola's equation.

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

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

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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=2 sin 8t, y=2 cos 8t

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37–48. Polar-to-Cartesian coordinates Convert the following equations to Cartesian coordinates. Describe the resulting curve.

r = 6 cos θ + 8 sin θ

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