Ch.12 - Parametric and Polar Curves

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch.12 - Parametric and Polar Curves

Problem 12.R.35

Chapter 12, Problem 12.R.35

Polar conversion Write the equation r ² +r(2sinθ−6cosθ)=0 in Cartesian coordinates and identify the corresponding curve.

Verified step by step guidance

Start with the given polar equation: \(r^{2} + r(2\sin\theta - 6\cos\theta) = 0\).

Recall the relationships between polar and Cartesian coordinates: \(x = r\cos\theta\), \(y = r\sin\theta\), and \(r^{2} = x^{2} + y^{2}\).

Rewrite each term in the equation using these relationships: replace \(r^{2}\) with \(x^{2} + y^{2}\), replace \(r\sin\theta\) with \(y\), and replace \(r\cos\theta\) with \(x\).

Substitute these into the equation to get: \((x^{2} + y^{2}) + (2y - 6x) = 0\).

Simplify and rearrange the equation to standard Cartesian form, then analyze the resulting equation to identify the type of curve it represents (such as a circle, ellipse, parabola, or hyperbola).

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Key Concepts

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

Polar to Cartesian Coordinate Conversion

This involves translating equations from polar form (r, θ) to Cartesian form (x, y) using the relationships x = r cos θ and y = r sin θ. Understanding these conversions allows one to rewrite polar equations in terms of x and y for easier analysis.

Algebraic Manipulation of Polar Equations

Rearranging and simplifying polar equations often requires substituting r and trigonometric terms with Cartesian equivalents and then algebraically manipulating the resulting expressions. This step is crucial to express the equation in a recognizable Cartesian form.

Identification of Conic Sections

Once the equation is in Cartesian form, recognizing the type of curve (circle, ellipse, parabola, or hyperbola) involves comparing it to standard conic section equations. This helps in understanding the geometric nature of the curve represented by the original polar equation.

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Related Practice

Textbook Question

51–52. {Use of Tech} Arc length of polar curves Find the approximate length of the following curves.

The limaçon r=3−6cosθ

Textbook Question

53–57. Conic sections

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

x = 16y²

Textbook Question

53–57. Conic sections

c. Find the eccentricity of the curve.

y² - 4x² = 16

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Textbook Question

A polar conic section Consider the equation r² = sec2θ

b. Find the vertices, foci, directrices, and eccentricity of the curve."

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Textbook Question

Parabola-hyperbola tangency: Let P be the parabola y = px² and H be the right half of the hyperbola x² - y² = 1.

a. For what value of p is P tangent to H?

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Textbook Question

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

a. The point with Cartesian coordinates (−2, 2) has polar coordinates (2√2, 3π/4), (2√2, 11π/4), (2√2, −5π/4), and (−2√2,−π/4).

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