Textbook Question
Convert each equation to standard form by completing the square on x and y. Then graph the ellipse and give the location of its foci. 25x²+4y² – 150x + 32y + 189 = 0
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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 55
Identify each equation without completing the square. 100x2 - 7y2 + 90y - 368 = 0
Verified step by step guidance1
Rewrite the given equation to group the terms involving \( y \) together: \( 100x^2 - 7y^2 + 90y - 368 = 0 \).
Focus on the quadratic terms: \( 100x^2 \) and \( -7y^2 \). Notice that the \( x^2 \) term is positive and the \( y^2 \) term is negative, which suggests the equation might represent a hyperbola.
Look at the linear term in \( y \), which is \( 90y \). This term shifts the center of the conic but does not change its fundamental type.
Recall the general forms of conic sections:
- Circle: \( Ax^2 + Ay^2 + ... = 0 \) with equal coefficients for \( x^2 \) and \( y^2 \)
- Ellipse: \( Ax^2 + By^2 + ... = 0 \) with \( A \) and \( B \) positive and unequal
- Hyperbola: \( Ax^2 - By^2 + ... = 0 \) or \( -Ax^2 + By^2 + ... = 0 \) with opposite signs for \( x^2 \) and \( y^2 \)
- Parabola: only one squared term
Based on the signs and coefficients of the squared terms, identify the conic as a hyperbola without completing the square.
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3mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Types of Conic Sections
Conic sections are curves obtained by intersecting a plane with a cone, including circles, ellipses, parabolas, and hyperbolas. Each type has a distinct general equation form, and identifying the conic involves analyzing the coefficients of the squared terms and their signs.
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Geometries from Conic Sections
General Form of a Conic Equation
The general second-degree equation Ax² + Bxy + Cy² + Dx + Ey + F = 0 represents conic sections. By examining coefficients A and C, and the presence or absence of the Bxy term, one can classify the conic without completing the square.
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Discriminant of a Conic Section
The discriminant, given by B² - 4AC, helps identify the conic type: if it is less than zero, the conic is an ellipse or circle; if zero, a parabola; and if greater than zero, a hyperbola. This method allows classification without rewriting the equation.
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Related Practice
Textbook Question
In Exercises 51–56, graph each relation. Use the relation's graph to determine its domain and range.
Textbook Question
Identify each equation without completing the square.
Textbook Question
Convert each equation to standard form by completing the square on x and y. Then graph the ellipse and give the location of its foci. 4x² + y²+ 16x - 6y - 39 = 0
Textbook Question
Convert each equation to standard form by completing the square on x and y. Then graph the ellipse and give the location of its foci. 9x² + 16y² – 18x + 64y – 71 = 0
Textbook Question
Identify each equation without completing the square.