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Ch.12 - Parametric and Polar Curves
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh.12 - Parametric and Polar CurvesProblem 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. 
Graph of a hyperbola centered at the origin with labeled vertices at (±4, 0) and foci at (±5, 0).

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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.
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Graph Hyperbolas NOT at the Origin

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.
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Foci and Vertices of Hyperbolas

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

73–76. Tangent lines Find an equation of the line tangent to the curve at the point corresponding to the given value of t.


x=cos t+t sin t,y=sin t−t cos t; t=π/4

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Textbook Question

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


r = 3

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Textbook Question

77–80. Slopes of tangent lines Find all points at which the following curves have the given slope.


x = 4 cos t, y = 4 sin t; slope = 1/2

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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 = cos t, y = 1 + sin t; 0 ≤ t ≤ 2π

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Textbook Question

69–72. Tangent lines Find an equation of the line tangent to the following curves at the given point.

x² = -6y; (-6, -6)

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Textbook Question

Plot the points with polar coordinates (2, π/6) and (−3, −π/2). Give two alternative sets of coordinate pairs for both points.

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