Domanda del libro di testo
Find the standard form of the equation of each hyperbola satisfying the given conditions. Center: (4, −2); Focus: (7, −2); vertex: (6, −2)
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Ch. 7 - Conic Sections
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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TOC.tableOfContents
- 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
Capitolo 8, Problema 11
Graph each ellipse and locate the foci. x² = 1 – 4y²
Guida verificata passo dopo passo1
Rewrite the given equation \(x^2 = 1 - 4y^2\) to the standard form of an ellipse equation. Start by moving all terms to one side: \(x^2 + 4y^2 = 1\).
Divide both sides of the equation by 1 to express it in the form \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\). Here, it becomes \(\frac{x^2}{1} + \frac{y^2}{\frac{1}{4}} = 1\).
Identify the values of \(a^2\) and \(b^2\). From the equation, \(a^2 = 1\) and \(b^2 = \frac{1}{4}\). Since \(a^2 > b^2\), the major axis is along the x-axis.
Calculate the focal distance \(c\) using the relationship \(c^2 = a^2 - b^2\). Substitute the values to get \(c^2 = 1 - \frac{1}{4}\).
Locate the foci at points \((\pm c, 0)\) on the x-axis. These points represent the foci of the ellipse. Finally, sketch the ellipse using the intercepts and foci.
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Standard Form of an Ellipse
An ellipse is typically expressed in the form \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \), where \((h,k)\) is the center, and \(a\) and \(b\) are the lengths of the semi-major and semi-minor axes. Understanding how to rewrite the given equation into this form is essential for graphing the ellipse.
Video consigliato:
5:12
Graph Ellipses at Origin
Identifying the Orientation of the Ellipse
The ellipse can be oriented horizontally or vertically depending on whether the \(x^2\) or \(y^2\) term is associated with the larger denominator. Recognizing the orientation helps in correctly plotting the ellipse and locating its axes.
Video consigliato:
4:50Graph Ellipses NOT at Origin
Locating the Foci of an Ellipse
The foci are two fixed points inside the ellipse, located along the major axis, found using \( c^2 = |a^2 - b^2| \). Knowing how to calculate \(c\) and place the foci relative to the center is crucial for completing the graph accurately.
Video consigliato:
5:30Foci and Vertices of an Ellipse
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