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Ch. 11 - Parametric Equations and Polar Coordinates
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 11 - Parametric Equations and Polar CoordinatesProblem 11.6.62
Chapter 11, Problem 11.6.62

Shifting Conic Sections


Find the center, foci, vertices, asymptotes, and radius, as appropriate, of the conic sections in Exercises 57-68.


9x² + 6y² + 36y = 0

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Rewrite the given equation to group the x and y terms: \(9x^{2} + 6y^{2} + 36y = 0\).
Complete the square for the y-terms to express the equation in standard form. First, factor out the coefficient of \(y^{2}\) from the y-terms: \(6(y^{2} + 6y)\). Then complete the square inside the parentheses by adding and subtracting the appropriate constant.
After completing the square, rewrite the equation in the form \(Ax^{2} + B(y - k)^{2} = C\), which will help identify the type of conic section (ellipse, hyperbola, or circle).
Divide through by the constant on the right side to normalize the equation to standard form, such as \(\frac{(x - h)^{2}}{a^{2}} + \frac{(y - k)^{2}}{b^{2}} = 1\) for an ellipse or \(\frac{(x - h)^{2}}{a^{2}} - \frac{(y - k)^{2}}{b^{2}} = 1\) for a hyperbola.
From the standard form, identify the center \((h, k)\), calculate the vertices and foci using the values of \(a\) and \(b\), find the equations of the asymptotes if it is a hyperbola, and determine the radius if it is a circle.

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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 the conic's center or vertex. It involves adding and subtracting terms to create perfect square trinomials, which simplifies identifying key features like the center or radius.
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Completing the Square to Rewrite the Integrand

Standard Forms of Conic Sections

Each conic section (circle, ellipse, parabola, hyperbola) has a standard equation form that highlights its geometric properties. Recognizing and converting the given equation into one of these forms helps determine the center, foci, vertices, asymptotes, and radius.
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Properties of Ellipses and Circles

Understanding the definitions and properties of ellipses and circles, such as the relationship between the center, foci, vertices, and radius, is essential. These properties guide the identification of key points and distances once the conic is in standard form.
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Related Practice
Textbook Question

Implicitly Defined Parametrizations


Assuming that the equations in Exercises 15−20 define x and y implicitly as differentiable functions x=f(t), y=g(t), find the slope of the curve x=f(t), y=g(t) at the given value of t.


x sin t + 2x = t, t sin t − 2t = y, t = π

Textbook Question

Finding Polar Areas


Find the areas of the regions in Exercises 9–18.


Inside the circle r = 4 sin θ and below the horizontal line r = 3 csc θ

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Textbook Question

Theory and Examples


Tangents Find equations for the tangents to the circle (x − 2)² + (y − 1)² = 5 at the points where the circle crosses the coordinate axes.

Textbook Question

Finding Cartesian from Parametric Equations


Exercises 1–18 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. (The graphs will vary with the equation used.) Indicate the portion of the graph traced by the particle and the direction of motion.


x = 2 sinh t, y = 2 cosh t, −∞<t<∞

Textbook Question

Circles


Sketch the circles in Exercises 53–56. Give polar coordinates for their centers and identify their radii.


r = −2 cos θ

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Textbook Question

Ellipses


Exercises 25 and 26 give information about the foci and vertices of ellipses centered at the origin of the xy−plane. In each case, find the ellipse's standard−form equation from the given information.


Foci: ( ±√2, 0) Vertices: (±2,0)