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.70

Chapter 12, Problem 12.R.70

Conic parameters: A hyperbola has eccentricity e = 2 and foci (0, ±2). Find the location of the vertices and directrices.

Verified step by step guidance

Identify the given parameters: eccentricity \(e = 2\) and foci at \((0, \pm 2)\). Since the foci lie on the y-axis, the hyperbola is vertical with center at the origin \((0,0)\).

Recall the relationship between the center, foci, and vertices for a vertical hyperbola: the foci are at \((0, \pm c)\) and the vertices are at \((0, \pm a)\), where \(c\) is the focal distance and \(a\) is the distance from the center to each vertex.

From the foci coordinates, determine \(c\): since the foci are at \((0, \pm 2)\), we have \(c = 2\).

Use the eccentricity formula for a hyperbola: \(e = \frac{c}{a}\). Given \(e = 2\) and \(c = 2\), solve for \(a\) by rearranging to \(a = \frac{c}{e}\).

Find the directrices using the formula for vertical hyperbolas: the directrices are horizontal lines given by \(y = \pm \frac{a}{e}\). Substitute the values of \(a\) and \(e\) to find their locations.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

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

Eccentricity of a Hyperbola

Eccentricity (e) measures how much a conic section deviates from being circular. For a hyperbola, e > 1, and it relates the distances between the foci and vertices. It is defined as e = c/a, where c is the distance from the center to a focus, and a is the distance from the center to a vertex.

Standard Form and Parameters of a Hyperbola

A hyperbola centered at the origin with vertical transverse axis has foci at (0, ±c) and vertices at (0, ±a). The relationship c² = a² + b² holds, where b relates to the conjugate axis. Knowing c and e allows calculation of a and b, which determine the shape and key points of the hyperbola.

Directrices of a Hyperbola

Directrices are fixed lines used to define a hyperbola via eccentricity. For a hyperbola with vertical transverse axis, the directrices are horizontal lines located at y = ±(a/e). They help in understanding the geometric definition and are essential for graphing and analyzing the hyperbola.

Recommended video:

5:50

Asymptotes of Hyperbolas

Related Practice

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?

views

Textbook Question

53–57. Conic sections

b. Use analytical methods to determine the location of the foci, vertices, and directrices.

x² - y²/2 = 1

Textbook Question

10–12. Parametric curves

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

x = t² + 4, y = -t, for -2 < t < 0; (5, 1)

Textbook Question

53–57. Conic sections

b. Use analytical methods to determine the location of the foci, vertices, and directrices.

4x² + 8y² = 16

Textbook Question

Cartesian conversion Write the equation x=y ² in polar coordinates and state values of θ that produce the entire graph of the parabola.

views

Textbook Question

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

b. At what point does the tangency occur?

views