Domanda del libro di testo
Use vertices and asymptotes to graph each hyperbola. Locate the foci and find the equations of the asymptotes. 4y2−x2=1
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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 17
Graph each ellipse and locate the foci. 7x² = 35-5y²
Guida verificata passo dopo passo1
Rewrite the given equation to standard form of an ellipse. Start with the equation: \(7x^{2} = 35 - 5y^{2}\). Move all terms to one side to get \(7x^{2} + 5y^{2} = 35\).
Divide every term by 35 to normalize the equation: \(\frac{7x^{2}}{35} + \frac{5y^{2}}{35} = \frac{35}{35}\), which simplifies to \(\frac{x^{2}}{5} + \frac{y^{2}}{7} = 1\).
Identify the values of \(a^{2}\) and \(b^{2}\) from the standard form \(\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1\). Here, \(a^{2} = 5\) and \(b^{2} = 7\). Since \(b^{2} > a^{2}\), the major axis is vertical.
Calculate the focal distance \(c\) using the formula \(c^{2} = b^{2} - a^{2}\). Substitute the values to get \(c^{2} = 7 - 5\).
Locate the foci on the graph along the major axis (the y-axis) at points \((0, \pm c)\). Then, sketch the ellipse centered at the origin with vertices at \((0, \pm \sqrt{7})\) and co-vertices at \((\pm \sqrt{5}, 0)\).
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Standard Form of an Ellipse
An ellipse can be expressed in the standard form as (x-h)²/a² + (y-k)²/b² = 1, where (h, k) is the center, and a and b are the lengths of the semi-major and semi-minor axes. Converting the given equation into this form is essential for graphing and identifying key features.
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5:12
Graph Ellipses at Origin
Identifying the Center and Axes Lengths
After rewriting the ellipse equation in standard form, determine the center coordinates and the values of a and b. These values represent the distances from the center to the ellipse's vertices along the major and minor axes, which are crucial for accurate graphing.
Video consigliato:
05:01Identifying Intervals of Unknown Behavior
Locating the Foci of an Ellipse
The foci are two fixed points inside the ellipse, located along the major axis, defined by the distance c from the center, where c² = |a² - b²|. Knowing how to calculate and plot the foci helps in understanding the ellipse's geometric properties.
Video consigliato:
5:30Foci and Vertices of an Ellipse
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