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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.R.7a
Chapter 12, Problem 12.R.7a

7–8. Parametric curves and tangent lines
a. Eliminate the parameter to obtain an equation in x and y.
x = 8cos t + 1, y = 8sin t + 2, for 0 ≤ t ≤ 2π; t = π/3

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Identify the given parametric equations: \(x = 8\cos t + 1\) and \(y = 8\sin t + 2\) with the parameter \(t\) in the interval \(0 \leq t \leq 2\pi\).
Recall the Pythagorean identity: \(\cos^2 t + \sin^2 t = 1\). This will help us eliminate the parameter \(t\) by expressing \(\cos t\) and \(\sin t\) in terms of \(x\) and \(y\).
Isolate \(\cos t\) and \(\sin t\) from the parametric equations: \(\cos t = \frac{x - 1}{8}\) and \(\sin t = \frac{y - 2}{8}\).
Substitute these expressions into the Pythagorean identity to get an equation involving only \(x\) and \(y\): \(\left(\frac{x - 1}{8}\right)^2 + \left(\frac{y - 2}{8}\right)^2 = 1\).
Simplify the equation by multiplying both sides by \(64\) (since \(8^2 = 64\)) to obtain the Cartesian equation of the curve without the parameter \(t\).

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Key Concepts

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

Parametric Equations

Parametric equations express the coordinates of points on a curve as functions of a parameter, often denoted as t. Instead of y as a function of x, both x and y depend on t, allowing the description of more complex curves like circles or ellipses.
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Parameterizing Equations

Eliminating the Parameter

Eliminating the parameter involves manipulating the parametric equations to remove t, resulting in a direct relationship between x and y. This often requires using trigonometric identities or algebraic techniques to rewrite the curve in Cartesian form.
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Eliminating the Parameter

Tangent Lines to Parametric Curves

The tangent line to a parametric curve at a given parameter value t is found by computing derivatives dx/dt and dy/dt, then using dy/dx = (dy/dt)/(dx/dt). This slope helps write the equation of the tangent line at the specified point on the curve.
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Differentiation of Parametric Curves
Related Practice
Textbook Question

General equations for a circle Prove that the equations  

X = a cos t + b sin t,  y = c cos t + d sin t  

where a, b, c, and d are real numbers, describe a circle of radius R provided a² +c² =b² +d² = R² and ab+cd=0. 

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Textbook Question

58–59. Tangent lines Find an equation of the line tangent to the following curves at the given point. Check your work with a graphing utility.

x²/16 - y²/9 = 1; (20/3, -4)

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Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

e. The hyperbola y²/2 - x²/4 = 1 has no x-intercept.

Textbook Question

53–57. Conic sections

d. Make an accurate graph of the curve.

x = 16y²

Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

c. The polar coordinates (3, -3π/4) and (-3, π/4) describe the same point in the plane.

Textbook Question

53–57. Conic sections

a. Determine whether the following equations describe a parabola, an ellipse, or a hyperbola.

x²/4 + y²/25 = 1