Textbook Question
Find the standard form of the equation of the ellipse satisfying the given conditions. Major axis horizontal with length 12; length of minor axis = 4; center: (-3,5)
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Ch. 7 - Conic Sections
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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Table of contents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Chapter 8, Problem 5
Find the focus and directrix of the parabola with the given equation. Then graph the parabola.
Verified step by step guidance1
Recognize that the given equation \(y^2 = 16x\) is a parabola that opens horizontally because the \(y\) variable is squared and \(x\) is to the first power.
Rewrite the equation in the standard form of a horizontal parabola: \(y^2 = 4px\). Here, \(4p = 16\), so solve for \(p\) by dividing both sides by 4, giving \(p = 4\).
Identify the vertex of the parabola, which is at the origin \((0,0)\) since the equation is not shifted.
Determine the focus using the value of \(p\). For a parabola that opens to the right, the focus is at \((p, 0)\), so substitute \(p = 4\) to get the focus at \((4, 0)\).
Find the directrix, which is a vertical line given by \(x = -p\). Substitute \(p = 4\) to get the directrix equation \(x = -4\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Standard Form of a Parabola
The standard form of a parabola's equation helps identify its orientation and key features. For a parabola opening horizontally, the form is (y - k)^2 = 4p(x - h), where (h, k) is the vertex and p determines the distance to the focus and directrix.
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Parabolas as Conic Sections
Focus and Directrix of a Parabola
The focus is a fixed point inside the parabola, and the directrix is a line outside it. The parabola consists of points equidistant from the focus and directrix. The value p in the equation gives the distance from the vertex to both the focus and directrix.
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Graphing a Parabola
Graphing involves plotting the vertex, focus, and directrix, then sketching the curve that is symmetric about the axis through the vertex and focus. Knowing the orientation (horizontal or vertical) and distances helps accurately draw the parabola.
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Related Practice
Textbook Question
Find the standard form of the equation of each hyperbola satisfying the given conditions. Foci: (0, −3), (0, 3) ; vertices: (0, −1), (0, 1)
Textbook Question
Find the standard form of the equation of each hyperbola satisfying the given conditions. Foci: (−4, 0), (4, 0); vertices:(−3, 0), (3, 0)
Textbook Question
Graph each ellipse and locate the foci. x2/25 +y2/64 = 1
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Textbook Question
Find the focus and directrix of the parabola with the given equation. Then graph the parabola. y2 = - 8x
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Textbook Question
Graph each ellipse and locate the foci. x2/49 +y2/81 = 1