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Ch.12 - Parametric and Polar Curves
Briggs - Calculus: Early Transcendentals 3rd Edition
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Tutti i libri di testoBriggs 3rd EditionCh.12 - Parametric and Polar CurvesProblema 12.R.60b
Capitolo 12, Problema 12.R.60b

A polar conic section Consider the equation r² = sec2θ
b. Find the vertices, foci, directrices, and eccentricity of the curve."

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Rewrite the given polar equation \(r^2 = \sec^2 \theta\) in a more standard form. Recall that \(\sec \theta = \frac{1}{\cos \theta}\), so express \(r\) in terms of \(\theta\) and simplify accordingly.
Identify the type of conic by comparing the equation to the general polar conic form \(r = \frac{ed}{1 \pm e \cos \theta}\) or \(r = \frac{ed}{1 \pm e \sin \theta}\). This will help determine the eccentricity \(e\) and the directrix distance \(d\).
Find the eccentricity \(e\) by analyzing the relationship between \(r\) and \(\theta\) in the equation. Recall that eccentricity measures how much the conic deviates from being a circle.
Determine the vertices by finding the values of \(r\) when \(\theta\) corresponds to the maximum and minimum distances from the pole (origin). This usually happens when the denominator in the standard form is minimized or maximized.
Locate the foci and directrices using the eccentricity and the directrix distance. The focus is at the pole for polar conics, and the directrix lines are located at distances related to \(d\) along the polar axis.

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Polar Coordinates and Polar Equations

Polar coordinates represent points using a radius r and angle θ from the origin. Polar equations express curves in terms of r and θ, allowing the description of conic sections like circles, ellipses, parabolas, and hyperbolas in a coordinate system centered at the pole.
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Intro to Polar Coordinates

Conic Sections in Polar Form

Conic sections can be represented in polar form using equations involving r, θ, eccentricity e, and directrix. The general form r = ed / (1 ± e cos θ) or r = ed / (1 ± e sin θ) relates the radius to the angle, eccentricity, and directrix, helping identify vertices, foci, and directrices.
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Parabolas as Conic Sections

Eccentricity, Vertices, Foci, and Directrices

Eccentricity (e) measures how much a conic deviates from a circle; e=0 is a circle, 0<e<1 an ellipse, e=1 a parabola, and e>1 a hyperbola. Vertices are points closest or farthest from the pole, foci are fixed points defining the conic, and directrices are lines used to define the conic via the eccentricity.
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Foci and Vertices of an Ellipse
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