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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

9–13. Graph the points with the following polar coordinates. Give two alternative representations of the points in polar coordinates.


(-4, 3π/2)

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.

The length of the latus rectum of a hyperbola centered at the origin is (2b²)/a = 2b√(1 - e²)

Textbook Question

37–48. Polar-to-Cartesian coordinates Convert the following equations to Cartesian coordinates. Describe the resulting curve.


r cos θ = -4

Textbook Question

39–50. Equations of ellipses and hyperbolas Find an equation of the following ellipses and hyperbolas, assuming the center is at the origin. 


A hyperbola with vertices (±4, 0) and foci (±6, 0)

Textbook Question

39–50. Equations of ellipses and hyperbolas Find an equation of the following ellipses and hyperbolas, assuming the center is at the origin. 

Textbook Question

15–22. Sets in polar coordinates Sketch the following sets of points.


r = 3