Textbook Question
53–57. Conic sections
b. Use analytical methods to determine the location of the foci, vertices, and directrices.
x² - y²/2 = 1
Ch.12 - Parametric and Polar Curves
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
Not the one you use?Change textbookSelect a different textbook
Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 12, Problem 12.R.10a
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)
Verified step by step guidance1
Identify the given parametric equations: \(x = t^{2} + 4\) and \(y = -t\), with the parameter \(t\) in the interval \(-2 < t < 0\).
Express the parameter \(t\) in terms of \(y\) from the second equation: since \(y = -t\), then \(t = -y\).
Substitute \(t = -y\) into the first equation to eliminate the parameter: \(x = (-y)^{2} + 4\).
Simplify the expression: \(x = y^{2} + 4\).
Write the final Cartesian equation relating \(x\) and \(y\): \(x = y^{2} + 4\), which represents the curve without the parameter \(t\).
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
1mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Parametric Equations
Parametric equations express the coordinates of points on a curve as functions of a parameter, often denoted as t. Instead of y as a function of x, both x and y depend on t, allowing the description of more complex curves.
Recommended video:
Guided course
08:02
Parameterizing Equations
Eliminating the Parameter
Eliminating the parameter involves manipulating the parametric equations to remove t, resulting in a direct relationship between x and y. This often requires solving one equation for t and substituting into the other to find an explicit or implicit equation of the curve.
Recommended video:
Guided course
05:59Eliminating the Parameter
Domain Restrictions on the Parameter
The parameter t is often restricted to a specific interval, which limits the portion of the curve described. Understanding these bounds is essential to correctly interpret the curve's segment and verify points like (5, 1) lie within the given parameter range.
Recommended video:
Guided course
05:59Eliminating the Parameter
Related Practice
Textbook Question
Conic parameters: A hyperbola has eccentricity e = 2 and foci (0, ±2). Find the location of the vertices and directrices.
1
views
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.
1
views
Textbook Question
14–18. Parametric descriptions Write parametric equations for the following curves. Solutions are not unique.
The segment of the curve x=y ³ +y+1 that starts at (1, 0) and ends at (11, 2).
1
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?
1
views