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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 12.4.33
Chapter 12, Problem 12.4.33

31–38. Equations of parabolas Find an equation of the following parabolas. Unless otherwise specified, assume the vertex is at the origin.
A parabola with focus at (3, 0)

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1
Recall that a parabola with its vertex at the origin and focus on the x-axis has the standard form equation \(y^2 = 4px\), where \(p\) is the distance from the vertex to the focus.
Identify the value of \(p\) from the given focus. Since the focus is at \((3, 0)\), the distance \(p\) is 3.
Substitute \(p = 3\) into the standard form equation to get \(y^2 = 4 \times 3 \times x\).
Simplify the equation to \(y^2 = 12x\), which represents the parabola with vertex at the origin and focus at \((3, 0)\).
Verify the orientation: since the focus is on the positive x-axis, the parabola opens to the right, consistent with the equation \(y^2 = 4px\) where \(p > 0\).

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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. Understanding this geometric definition is essential to derive the equation of a parabola given its focus and vertex.
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Standard Form of a Parabola with Vertex at the Origin

When the vertex is at the origin, a parabola with a horizontal axis of symmetry has the form x² = 4py (vertical) or y² = 4px (horizontal). The parameter p represents the distance from the vertex to the focus, which helps in writing the equation.
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Relationship Between Focus and Parameter p

The distance p is the distance from the vertex to the focus along the axis of symmetry. For a focus at (3, 0), p = 3, indicating the parabola opens horizontally. This value is used directly in the standard form equation y² = 4px.
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Related Practice
Textbook Question

Tangent lines for a hyperbola Find an equation of the line tangent to the hyperbola x²/a² + y²/b² = 1 at the point (x₀, y₀)

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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 = 4 + sin θ; (4, 0) and (3, 3π/2)

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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)

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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.

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

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Textbook Question

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


r cos θ = -4

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Textbook Question

37–52. Curves to parametric equations Find parametric equations for the following curves. Include an interval for the parameter values. Answers are not unique.


The left half of the parabola y=x ² +1, originating at (0, 1)

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