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Ch. 5 - Complex Numbers, Polar Coordinates and Parametric Equations
Blitzer - Trigonometry 3rd Edition
Blitzer3rd EditionTrigonometryISBN: 9780137316601Not the one you use?Change textbook
All textbooksBlitzer 3rd EditionCh. 5 - Complex Numbers, Polar Coordinates and Parametric EquationsProblem 5.5.49
Chapter 5, Problem 5.5.49

In Exercises 45–52, use your answers from Exercises 41–44 and the parametric equations given in Exercises 41–44 to find a set of parametric equations for the conic section or the line.


Hyperbola: Vertices: (4,0) and (−4,0); Foci: (6,0) and (−6,0)

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Identify the center of the hyperbola by finding the midpoint of the vertices. Since the vertices are at (4,0) and (-4,0), the center is at \(\left( \frac{4 + (-4)}{2}, \frac{0 + 0}{2} \right) = (0,0)\).
Determine the distance from the center to each vertex, which is the value of \(a\). Here, \(a = 4\) because the vertices are 4 units away from the center along the x-axis.
Determine the distance from the center to each focus, which is the value of \(c\). Here, \(c = 6\) because the foci are 6 units away from the center along the x-axis.
Calculate \(b\) using the relationship for hyperbolas: \(c^2 = a^2 + b^2\). Rearranged, this is \(b = \sqrt{c^2 - a^2}\). Substitute the values of \(a\) and \(c\) to find \(b\).
Write the parametric equations for the hyperbola centered at the origin with a horizontal transverse axis: \(x = a \sec t\) \(y = b \tan t\) where \(t\) is the parameter.

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Key Concepts

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

Parametric Equations of Conic Sections

Parametric equations express the coordinates of points on a curve as functions of a parameter, often denoted t. For conic sections like hyperbolas, these equations allow a clear representation of the curve's shape and position, facilitating analysis and graphing.
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Properties of a Hyperbola

A hyperbola is defined by its vertices and foci, which determine its shape and orientation. The distance between vertices gives the transverse axis length, while the foci relate to the hyperbola's eccentricity, crucial for forming its standard equation and parametric form.
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Relationship Between Vertices, Foci, and Parameters a, b, c

In a hyperbola, a represents the distance from the center to each vertex, c the distance to each focus, and b relates to the conjugate axis. These parameters satisfy the equation c² = a² + b², which is essential for deriving the hyperbola's parametric equations.
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Related Practice
Textbook Question

In Exercises 45–52, find the quotient z₁/z₂ of the complex numbers. Leave answers in polar form. In Exercises 49–50, express the argument as an angle between 0° and 360°.

z₁ = 20(cos 75° + i sin 75°)

z₂ = 4(cos 25° + i sin 25°)

Textbook Question

In Exercises 41–43, eliminate the parameter. Write the resulting equation in standard form.


A hyperbola: x = h + a sec t, y = k + b tan t

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Textbook Question

Convert each rectangular equation to a polar equation that expresses r in terms of θ.

y = 3

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Textbook Question

In Exercises 53–64, use DeMoivre's Theorem to find the indicated power of the complex number. Write answers in rectangular form. [√2 (cos (5π/6) + i sin (5π/6))]⁴

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Textbook Question

In Exercises 53–64, use DeMoivre's Theorem to find the indicated power of the complex number. Write answers in rectangular form. (1 − i)⁵

Textbook Question

In Exercises 45–52, use your answers from Exercises 41–44 and the parametric equations given in Exercises 41–44 to find a set of parametric equations for the conic section or the line.


Circle: Center: (3,5); Radius: 6

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