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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 12.4.41
Chapter 12, Problem 12.4.41

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)

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Identify the type of conic: Since the vertices and foci lie on the x-axis, the hyperbola opens left and right, so its standard form is \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\).
Determine the value of \(a\): The vertices are at \((\pm 4, 0)\), so \(a = 4\) and therefore \(a^2 = 16\).
Determine the value of \(c\): The foci are at \((\pm 6, 0)\), so \(c = 6\) and therefore \(c^2 = 36\).
Use the relationship between \(a\), \(b\), and \(c\) for hyperbolas: \(c^2 = a^2 + b^2\). Substitute the known values to find \(b^2\).
Write the equation of the hyperbola using the values of \(a^2\) and \(b^2\): \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\).

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Key Concepts

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

Standard Form 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 conjugate axis. Understanding this form is essential to write the equation 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 \(a\) and \(c\) allows you to find \(b\), which is necessary to complete the hyperbola's equation.
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Foci and Vertices of Hyperbolas

Interpreting Coordinates of Vertices and Foci

Vertices and foci coordinates provide direct information about the hyperbola's orientation and size. Vertices at (±4, 0) indicate a horizontal transverse axis with \(a=4\), and foci at (±6, 0) give \(c=6\). This helps determine the parameters needed to write the hyperbola's equation.
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Foci and Vertices of an Ellipse
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

90–94. Focal chords A focal chord of a conic section is a line through a focus joining two points of the curve. The latus rectum is the focal chord perpendicular to the major axis of the conic. Prove the following properties.

The length of the latus rectum of a hyperbola centered at the origin is (2b²)/a = 2b√(1 - e²)

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. 

Textbook Question

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


r = 3