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Use hyperbolic functions to parametrize the intersection of the surfaces x² - y² = 4 and z = 5xy. (Use symbolic notation and fractions where needed. Use hyperbolic cosine for parametrization x variable.)

Use hyperbolic functions to parametrize the intersection of the surfaces x² - y² = 4 and z = 5xy. (Use symbolic notation and fractions where needed. Use hyperbolic cosine for parametrization x variable.)

Trigonometry (MindTap Course List)
Trigonometry (MindTap Course List)
10th Edition
ISBN:9781337278461
Author:Ron Larson
Publisher:Ron Larson
Chapter6: Topics In Analytic Geometry
Section6.6: Parametric Equations
Problem 5ECP: Write parametric equations for a cycloid traced by a point P on a circle of radius a as the circle...
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**Hyperbolic Functions and Parametric Equations** **Problem Statement:** Use hyperbolic functions to parametrize the intersection of the surfaces \(x^2 - y^2 = 4\) and \(z = 5xy\). (Use symbolic notation and fractions where needed. Use hyperbolic cosine for parametrization \(x\) variable.) **Equations:** \[x(t) = \quad \rule{20em}{0.5pt}\] \[y(t) = \quad \rule{20em}{0.5pt}\] \[z(t) = \quad \rule{20em}{0.5pt}\] **Instructions:** 1. **Identify the Parameterization Technique:** - For the variable \(x\), use the hyperbolic cosine function (\(\cosh(t)\)). - For the variable \(y\), use the hyperbolic sine function (\(\sinh(t)\)). 2. **Determine Parametric Equations:** - Substitute these hyperbolic functions into the equation \(x^2 - y^2 = 4\). - Ensure the equations satisfy both the given surfaces \(x^2 - y^2 = 4\) and \(z = 5xy\). 3. **Input the Parametric Equations:** - Write the correct parametric equations for \(x(t)\), \(y(t)\), and \(z(t)\) based on the parameter \(t\). **Note:** The hyperbolic functions \(\cosh(t)\) and \(\sinh(t)\) are defined as: \[ \cosh(t) = \frac{e^t + e^{-t}}{2} \] \[ \sinh(t) = \frac{e^t - e^{-t}}{2} \]
Transcribed Image Text:**Hyperbolic Functions and Parametric Equations** **Problem Statement:** Use hyperbolic functions to parametrize the intersection of the surfaces \(x^2 - y^2 = 4\) and \(z = 5xy\). (Use symbolic notation and fractions where needed. Use hyperbolic cosine for parametrization \(x\) variable.) **Equations:** \[x(t) = \quad \rule{20em}{0.5pt}\] \[y(t) = \quad \rule{20em}{0.5pt}\] \[z(t) = \quad \rule{20em}{0.5pt}\] **Instructions:** 1. **Identify the Parameterization Technique:** - For the variable \(x\), use the hyperbolic cosine function (\(\cosh(t)\)). - For the variable \(y\), use the hyperbolic sine function (\(\sinh(t)\)). 2. **Determine Parametric Equations:** - Substitute these hyperbolic functions into the equation \(x^2 - y^2 = 4\). - Ensure the equations satisfy both the given surfaces \(x^2 - y^2 = 4\) and \(z = 5xy\). 3. **Input the Parametric Equations:** - Write the correct parametric equations for \(x(t)\), \(y(t)\), and \(z(t)\) based on the parameter \(t\). **Note:** The hyperbolic functions \(\cosh(t)\) and \(\sinh(t)\) are defined as: \[ \cosh(t) = \frac{e^t + e^{-t}}{2} \] \[ \sinh(t) = \frac{e^t - e^{-t}}{2} \]
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