Skip to main content
Ch. 11 - Parametric Equations and Polar Coordinates
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoHass 15th EditionCh. 11 - Parametric Equations and Polar CoordinatesProblema 11.AAE.22
Capitolo 11, Problema 11.AAE.22

Polar Coordinates


Exercises 19–22 give the eccentricities of conic sections with one focus at the origin of the polar coordinate plane, along with the directrix for that focus. Find a polar equation for each conic section.


e = 1/3, r sin θ = −6

Guida verificata passo dopo passo
1
Recall the general polar form of a conic section with one focus at the origin: \[ r = \frac{ed}{1 + e \cos \theta} \quad \text{or} \quad r = \frac{ed}{1 + e \sin \theta} \] where \(e\) is the eccentricity and \(d\) is the distance from the focus to the directrix.
Identify the given eccentricity \(e = \frac{1}{3}\) and the directrix equation \(r \sin \theta = -6\). Since the directrix involves \(r \sin \theta\), it corresponds to a horizontal line in polar coordinates, so the conic equation will use the \(\sin \theta\) form.
Rewrite the directrix equation \(r \sin \theta = -6\) to find the distance \(d\) from the origin to the directrix. Note that \(r \sin \theta\) represents the \(y\)-coordinate in Cartesian coordinates, so the directrix is the line \(y = -6\). Thus, \(d = 6\) (distance is positive).
Substitute \(e = \frac{1}{3}\) and \(d = 6\) into the polar conic formula with \(\sin \theta\): \[ r = \frac{e d}{1 + e \sin \theta} = \frac{\frac{1}{3} \times 6}{1 + \frac{1}{3} \sin \theta} \]
Simplify the expression to write the final polar equation of the conic section in terms of \(r\) and \(\theta\).

Risposta video verificata per un problema simile:

Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.
Durata del video:
3m

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

Polar Coordinates and Polar Equations

Polar coordinates represent points using a radius and an angle (r, θ) from the origin. Polar equations express curves in terms of r and θ, which is essential for describing conic sections centered at a focus located at the origin.
Video consigliato:
05:32
Intro to Polar Coordinates

Eccentricity of Conic Sections

Eccentricity (e) measures how much a conic section deviates from being circular. For conics with a focus at the origin, e < 1 indicates an ellipse, e = 1 a parabola, and e > 1 a hyperbola. This value helps determine the shape and equation of the conic.
Video consigliato:
Percorso guidato
5:33
Parabolas as Conic Sections

Directrix in Polar Form

The directrix is a fixed line used to define conics relative to a focus. In polar coordinates, the directrix can be expressed as a linear equation involving r and θ, such as r sin θ = -6, which is used along with eccentricity to find the polar equation of the conic.
Video consigliato:
05:32
Intro to Polar Coordinates
Pratica correlata
Domanda del libro di testo

Finding Parametric Equations and Tangent Lines


Find parametric equations for the given curve.


9x² + 4y² = 36

18
views
Domanda del libro di testo

Graphing Conic Sections


Exercises 63-68 give equations for conic sections and tell how many units up or down and to the right or left each curve is to be shifted. Find an equation for the new conic section, and find the new foci, vertices, centers, and asymptotes, as appropriate. If the curve is a parabola, find the new directrix as well.


x²/169 + y²/144 = 1, right 5, up 12

33
views
Domanda del libro di testo

Hyperbolas and Eccentricity


In Exercises 17-24, find the eccentricity of the hyperbola. Then find and graph the hyperbola's foci and directrices.


y² − x² = 4

37
views
Domanda del libro di testo

Lines


Sketch the lines in Exercises 45–48 and find Cartesian equations for them.


r cos (θ + π/3) = 2

24
views
Domanda del libro di testo

Hyperbolas and Eccentricity


Exercises 25–28 give the eccentricities and the vertices or foci of hyperbolas centered at the origin of the xy-plane. In each case, find the hyperbola’s standard-form equation in Cartesian coordinates.


Eccentricity: 1.25

Foci: (0, ±5)

6
views
Domanda del libro di testo

Graphing Conic Sections


Find the eccentricities of the ellipses and hyperbolas in Exercises 59–62. Sketch each conic section. Include the foci, vertices, and asymptotes (as appropriate) in your sketch.


5y² − 4x² = 20

31
views