Ch. 11 - Parametric Equations and Polar Coordinates

Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook

Hass 15th Edition

Ch. 11 - Parametric Equations and Polar Coordinates

Problem 11.PE.76

Chapter 11, Problem 11.PE.76

Identifying Conic Sections

Complete the squares to identify the conic sections in Exercises 69-76. Find their foci, vertices, centers, and asymptotes (as appropriate). If the curve is a parabola, find its directrix as well.

x² + y² + 4x + 2y = 1

Verified step by step guidance

Start with the given equation: \(x^2 + y^2 + 4x + 2y = 1\).

Group the \(x\) terms and \(y\) terms together: \((x^2 + 4x) + (y^2 + 2y) = 1\).

Complete the square for each group: - For \(x^2 + 4x\), take half of 4 (which is 2), square it (which is 4), and add inside the parentheses. - For \(y^2 + 2y\), take half of 2 (which is 1), square it (which is 1), and add inside the parentheses.

Since you added \(4\) and \(1\) inside the equation, add the same amounts to the right side to keep the equation balanced: \(1 + 4 + 1\).

Rewrite the equation in standard form using the completed squares: \[(x + 2)^2 + (y + 1)^2 = \text{new constant}\]. This form represents a circle, so identify the center and radius from this equation.

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Key Concepts

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

Completing the Square

Completing the square is a method used to rewrite quadratic expressions in a form that reveals geometric properties. By adding and subtracting appropriate constants, you transform terms like x² + 4x into (x + 2)² - 4. This technique is essential for rewriting conic section equations into standard forms.

Identification of Conic Sections

Conic sections include circles, ellipses, parabolas, and hyperbolas, each defined by specific standard equations. Recognizing the form of the equation after completing the square helps determine the type of conic, such as a circle if x and y terms have equal coefficients and the same sign.

Key Features of Conic Sections

Each conic section has characteristic elements: centers and vertices for ellipses and hyperbolas, foci for all conics, asymptotes for hyperbolas, and directrices for parabolas. Finding these features involves using the standard form of the conic and applying formulas related to distances and axes.

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Related Practice

Textbook Question

Finding Parametric Equations and Tangent Lines

Find parametric equations for the given curve.

9x² + 4y² = 36

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Textbook Question

Identifying Parametric Equations in the Plane

Exercises 1–6 give parametric equations and parameter intervals for the motion of a particle in the xy-plane. Identify the particle’s path by finding a Cartesian equation for it. Graph the Cartesian equation and indicate the direction of motion and the portion traced by the particle.

x = √t, y = 1 − √t, t ≥ 0

Textbook Question

Identifying Parametric Equations in the Plane

x = 4 cos t, y = 9 sin t, 0 ≤ t ≤ 2π

Textbook Question

Area in Polar Coordinates

Find the areas of the regions in the polar coordinate plane described in Exercises 47–50.

Inside the cardioid r = 2(1 + sin θ) and outside the circle r = 2 sin θ

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Textbook Question

Polar to Cartesian Equations

Sketch the lines in Exercises 23-28. Also, find a Cartesian equation for each line.