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 12.4.49

Chapter 12, Problem 12.4.49

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 two branches opening left and right with vertices at (±4, 0) and foci at (±5, 0), this is a hyperbola centered at the origin with a horizontal transverse axis.

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

Determine the value of \(a\): The vertices are at (±4, 0), so \(a = 4\), which means \(a^2 = 16\).

Determine the value of \(c\): The foci are at (±5, 0), so \(c = 5\), which means \(c^2 = 25\).

Use the relationship between \(a\), \(b\), and \(c\) for hyperbolas: \(c^2 = a^2 + b^2\). Substitute the known values to solve for \(b^2\): \(25 = 16 + b^2\), then solve for \(b^2\).

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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 horizontal transverse axis has the equation \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \). Here, \(a\) is the distance from the center to each vertex along the x-axis, and \(b\) relates to the distance along the conjugate axis. Understanding this form is essential to write the equation from given vertices and foci.

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

For hyperbolas, \(a\) is the distance from the center to each vertex, and \(c\) is the distance to each focus. These satisfy the equation \( c^2 = a^2 + b^2 \). Knowing the coordinates of vertices and foci allows calculation of \(a\), \(c\), and subsequently \(b\), which are needed to form the hyperbola's equation.

Graph Interpretation and Coordinate Geometry

Interpreting the graph involves identifying key points such as vertices and foci and their coordinates. This helps in determining the values of \(a\) and \(c\) directly from the graph. Accurate reading of these points is crucial for applying the hyperbola formulas correctly.

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Intro to Polar Coordinates

Related Practice

Textbook Question

11–20. Slopes of tangent lines Find the slope of the line tangent to the following polar curves at the given points.

r = 4 + sin θ; (4, 0) and (3, 3π/2)

Textbook Question

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

r cos θ = -4

Textbook Question

31–38. Equations of parabolas Find an equation of the following parabolas. Unless otherwise specified, assume the vertex is at the origin.

A parabola that opens to the right with directrix x = -4

Textbook Question

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

A hyperbola with vertices (±4, 0) and foci (±6, 0)

Textbook Question

15–22. Sets in polar coordinates Sketch the following sets of points.

r = 3

Textbook Question

85–87. Grazing goat problems Consider the following sequence of problems related to grazing goats tied to a rope. (See the Guided Project Grazing goat problems.)

A circular corral of unit radius is enclosed by a fence. A goat inside the corral is tied to the fence with a rope of length 0≤a≤2 (see figure). What is the area of the region (inside the corral) that the goat can graze? Check your answer with the special cases a=0 and a=2.