Domanda del libro di testo
Graph each ellipse and give the location of its foci. (x +3)²/9 + (y -2)² = 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 43
Graph each ellipse and give the location of its foci. x²/25 + (y -2)² /36= 1
Guida verificata passo dopo passo1
Identify the standard form of the ellipse equation: \(\frac{x^2}{a^2} + \frac{(y - k)^2}{b^2} = 1\), where \((h, k)\) is the center of the ellipse. In this equation, \(h = 0\) and \(k = 3\).
Determine the values of \(a^2\) and \(b^2\) by comparing the denominators: \(a^2 = 9\) and \(b^2 = 49\). Since \(b^2 > a^2\), the major axis is vertical.
Find the lengths of the semi-major axis \(b = \sqrt{49} = 7\) and the semi-minor axis \(a = \sqrt{9} = 3\).
Locate the center of the ellipse at the point \((0, 3)\).
Calculate the distance \(c\) from the center to each focus using the formula \(c = \sqrt{b^2 - a^2} = \sqrt{49 - 9}\). The foci will be located along the major axis (vertical) at \((0, 3 \pm c)\).
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Standard Form of an Ellipse
An ellipse's equation in standard form is written as (x - h)²/a² + (y - k)²/b² = 1, where (h, k) is the center. The denominators a² and b² represent the squares of the ellipse's semi-major and semi-minor axes. Understanding this form helps identify the ellipse's size, shape, and position on the coordinate plane.
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5:12
Graph Ellipses at Origin
Identifying the Major and Minor Axes
The larger denominator between a² and b² determines the length of the major axis, while the smaller corresponds to the minor axis. The major axis is the longest diameter of the ellipse, and its orientation (horizontal or vertical) depends on whether a² or b² is larger. This distinction is crucial for graphing and locating the foci.
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05:01Identifying Intervals of Unknown Behavior
Finding 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 c allows you to find the exact coordinates of the foci, which are essential for understanding the ellipse's geometric properties and for accurate graphing.
Video consigliato:
5:30Foci and Vertices of an Ellipse
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