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Ch.12 - Parametric and Polar Curves
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh.12 - Parametric and Polar CurvesProblem 12.4.31
Chapter 12, Problem 12.4.31

31–38. Equations of parabolas Find an equation of the following parabolas. Unless otherwise specified, assume the vertex is at the origin.
A parabola that opens to the right with directrix x = -4

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1
Recall the definition of a parabola: it is the set of all points equidistant from the focus and the directrix.
Since the directrix is given as the vertical line \(x = -4\) and the parabola opens to the right, the axis of symmetry is horizontal along the x-axis.
The vertex is at the origin \((0,0)\), so the focus must be on the positive x-axis, at some point \((p,0)\), where \(p > 0\).
The distance from the vertex to the directrix is \(|p| = 4\), so the focus is at \((4,0)\).
Use the standard form of a parabola that opens right: \(y^2 = 4px\). Substitute \(p = 4\) to get the equation \(y^2 = 16x\).

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Key Concepts

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

Definition of a Parabola

A parabola is the set of all points equidistant from a fixed point called the focus and a fixed line called the directrix. This geometric definition helps derive the equation of the parabola by relating distances from any point on the curve to the focus and directrix.
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Orientation of Parabolas

The orientation of a parabola depends on the position of its focus and directrix. If the parabola opens right or left, its axis of symmetry is horizontal, and the equation involves x and y accordingly. For a parabola opening right, the directrix is vertical, and the equation typically has the form (y - k)^2 = 4p(x - h).
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Properties of Parabolas

Vertex at the Origin and Directrix

When the vertex is at the origin, the parabola's equation simplifies since h = 0 and k = 0. Given the directrix x = -4, the focus lies on the opposite side of the vertex at x = 4, allowing calculation of the parameter p, which determines the distance from the vertex to the focus or directrix and shapes the parabola's equation.
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Related Practice
Textbook Question

81–88. Arc length Find the arc length of the following curves on the given interval.


x = sin t, y = t - cos t; 0 ≤ t ≤ π/2

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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 = 8 + 2t, y = 1; −∞ < t < ∞

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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.

Let L be the latus rectum of the parabola y ² =4px for p>0. Let F be the focus of the parabola, P be any point on the parabola to the left of L, and D be the (shortest) distance between P and L. Show that for all P, D+|FP|+ is a constant. Find the constant.

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Textbook Question

Given three polar coordinate representations for the origin.

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Textbook Question

Second derivative Assume a curve is given by the parametric equations x=f(t) and y=g(t), where f and g are twice differentiable. Use the Chain Rule to show that y″x=(fʹ(t)g″(t)−gʹ(t)f″(t))/(fʹ(t))³.  

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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 = 1 - sin θ; (1/2, π/6)

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