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 12.4.18

Chapter 12, Problem 12.4.18

13–30. Graphing conic sections Determine whether the following equations describe a parabola, an ellipse, or a hyperbola, and then sketch a graph of the curve. For each parabola, specify the location of the focus and the equation of the directrix; for each ellipse, label the coordinates of the vertices and foci, and find the lengths of the major and minor axes; for each hyperbola, label the coordinates of the vertices and foci, and find the equations of the asymptotes.

4x = -y²

Verified step by step guidance

Rewrite the given equation \(4x = -y^2\) in a more standard form by isolating \(x\). This gives \(x = -\frac{1}{4} y^2\).

Recognize the form of the equation: since \(x\) is expressed as a quadratic function of \(y\), this is the equation of a parabola that opens either left or right.

Identify the vertex of the parabola. Because the equation is in the form \(x = a y^2 + h\), the vertex is at the point \((0,0)\).

Determine the direction the parabola opens. Since the coefficient of \(y^2\) is negative (\(-\frac{1}{4}\)), the parabola opens to the left along the \(x\)-axis.

Find the focus and directrix using the standard parabola formula \(x = \frac{1}{4p} y^2\). Here, \(\frac{1}{4p} = -\frac{1}{4}\), so solve for \(p\), then use \(p\) to locate the focus at \((h+p, k)\) and the directrix at \(x = h - p\).

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Key Concepts

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

Identification of Conic Sections from Equations

Conic sections are curves obtained by intersecting a plane with a cone, resulting in parabolas, ellipses, or hyperbolas. Recognizing the type involves rewriting the equation in a standard form and analyzing the degree and signs of variables. For example, an equation with one squared term and one linear term often represents a parabola.

Properties and Features of Parabolas

A parabola is defined as the set of points equidistant from a fixed point (focus) and a line (directrix). Its graph is symmetric about an axis. Key features include the vertex, focus, directrix, and axis of symmetry, which can be found by rewriting the equation in vertex form or standard form.

Graphing and Analyzing Conic Sections

Graphing conics requires identifying key points such as vertices, foci, and asymptotes (for hyperbolas). For parabolas, locating the focus and directrix helps in sketching. Understanding how to extract these features from the equation enables accurate graphing and interpretation of the curve's shape and orientation.

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Parabolas as Conic Sections

Related Practice

Textbook Question

11–20. Slopes of tangent lines Find the slope of the line tangent to the following polar curves at the given points.

r = 4 cos 2θ; at the tips of the leaves

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Textbook Question

33–40. Areas of regions Make a sketch of the region and its bounding curves. Find the area of the region.

The region inside the limaçon r = 2 + cos θ

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Textbook Question

15–30. Working with parametric equations Consider the following parametric equations.

a. Eliminate the parameter to obtain an equation in x and y.

b. Describe the curve and indicate the positive orientation.

x = 3 + t, y = 1 − t; 0 ≤ t ≤ 1

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Textbook Question

90–94. Focal chords A focal chord of a conic section is a line through a focus joining two points of the curve. The latus rectum is the focal chord perpendicular to the major axis of the conic. Prove the following properties.

The length of the latus rectum of the parabola y ² =4px or x ² =4py is 4|p|.

119

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Textbook Question

53–56. Circular motion Find parametric equations that describe the circular path of the following objects. For Exercises 53–55, assume (x, y) denotes the position of the object relative to the origin at the center of the circle. Use the units of time specified in the problem. There are many ways to describe any circle.

The tip of the 15-inch second hand of a clock completes one revolution in 60 seconds.

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Textbook Question

13–30. Graphing conic sections Determine whether the following equations describe a parabola, an ellipse, or a hyperbola, and then sketch a graph of the curve. For each parabola, specify the location of the focus and the equation of the directrix; for each ellipse, label the coordinates of the vertices and foci, and find the lengths of the major and minor axes; for each hyperbola, label the coordinates of the vertices and foci, and find the equations of the asymptotes.

x² + y²/9 = 1

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