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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 11.6.44a
Chapter 11, Problem 11.6.44a

Shifting Conic Sections


You may wish to review Section 1.2 before solving Exercises 39-56.


The hyperbola (y²/4) − (x²/5) = 1 is shifted 2 units down to generate the hyperbola (y + 2)²/4 − x²/5 = 1.
a. Find the center, foci, vertices, and asymptotes of the new hyperbola.

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Identify the original hyperbola equation: \(\frac{y^2}{4} - \frac{x^2}{5} = 1\). This is a vertical hyperbola centered at the origin \((0,0)\) because the \(y^2\) term is positive and comes first.
Recognize the shift applied: the hyperbola is shifted 2 units down, changing \(y\) to \(y + 2\). The new equation is \(\frac{(y + 2)^2}{4} - \frac{x^2}{5} = 1\). This means the center moves from \((0,0)\) to \((0, -2)\).
Find the center of the new hyperbola: since the shift is down by 2 units, the center is at \((0, -2)\).
Determine the vertices: for a vertical hyperbola of the form \(\frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1\), the vertices are located \(a\) units above and below the center. Here, \(a^2 = 4\), so \(a = 2\). Thus, vertices are at \((0, -2 + 2)\) and \((0, -2 - 2)\).
Find the foci: the distance \(c\) from the center satisfies \(c^2 = a^2 + b^2\). Given \(a^2 = 4\) and \(b^2 = 5\), calculate \(c = \sqrt{4 + 5}\). The foci are then at \((0, -2 + c)\) and \((0, -2 - c)\).
Write the equations of the asymptotes: for a vertical hyperbola centered at \((h,k)\), the asymptotes are given by \(y - k = \pm \frac{a}{b}(x - h)\). Substitute \(h=0\), \(k=-2\), \(a=2\), and \(b=\sqrt{5}\) to get the asymptote equations.

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Key Concepts

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

Shifting Conic Sections

Shifting a conic section involves translating its graph horizontally or vertically without changing its shape. For a hyperbola, shifting the center from (h, k) to (h, k + c) modifies the equation by replacing y with (y - (k + c)) or x with (x - (h + c)). This translation affects the location of key features like the center, vertices, and foci but not their relative distances.
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Properties of Hyperbolas

A hyperbola is defined by its center, vertices, foci, and asymptotes. The center is the midpoint between vertices; vertices lie along the transverse axis. The foci are points inside the hyperbola related to its eccentricity, and asymptotes are lines the hyperbola approaches at infinity. These features are derived from the standard form equation and parameters a, b, and c.
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Properties of Functions

Equations of Asymptotes for Hyperbolas

Asymptotes of a hyperbola are straight lines that the curve approaches but never touches. For a hyperbola centered at (h, k) with equation (y - k)²/a² - (x - h)²/b² = 1, the asymptotes have equations y - k = ±(a/b)(x - h). Understanding how to find and shift these lines is essential when the hyperbola is translated.
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Related Practice
Textbook Question

Polar Coordinates


Plot the following points, given in polar coordinates. Then find all the polar coordinates of each point.


a. (2, π/2)

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Textbook Question

Graphing Conic Sections


Sketch the parabolas in Exercises 55–58. Include the focus and directrix in each sketch.


y² = −(8/3)x

Textbook Question

Finding Parametric Equations


Find parametric equations and a parameter interval for the motion of a particle that starts at (a, 0) and traces the circle x²+y²=a².


a. once clockwise.


(There are many ways to do these, so your answers may not be the same as the ones at the back of the text.)

Textbook Question

Theory and Examples


Volume Find the volume of the solid generated by revolving the region enclosed by the ellipse 9x² + 4y² = 36 about the y−axis.

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Textbook Question

Cartesian to Polar Coordinates


Find the polar coordinates, 0 ≤ θ < 2π and r ≥ 0, of the following points given in Cartesian coordinates.


b. (-3,0)

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Textbook Question

Cycloid


a. Find the length of one arch of the cycloid x = a(t − sin t), y = a(1 − cos t).

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