Domanda del libro di testo
Convert each equation to standard form by completing the square on x or y. Then find the vertex, focus, and directrix of the parabola. Finally, graph the parabola. y2 - 2y + 12x - 35 = 0
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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 47
Graph each ellipse and give the location of its foci. (x − 1)²/2 + (y +3)² /5= 1
Guida verificata passo dopo passo1
Identify the standard form of the ellipse equation: \(\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1\), where \((h, k)\) is the center of the ellipse.
From the given equation \(\frac{(x - 2)^2}{7} + \frac{(y + 1)^2}{11} = 1\), determine the center of the ellipse as \((2, -1)\).
Compare the denominators to find \(a^2\) and \(b^2\). Since \(11 > 7\), set \(a^2 = 11\) and \(b^2 = 7\). This means the major axis is vertical.
Calculate the distance \(c\) from the center to each focus using the formula \(c = \sqrt{a^2 - b^2}\). Substitute the values to find \(c = \sqrt{11 - 7}\).
Locate the foci along the major axis (vertical axis) by adding and subtracting \(c\) from the \(y\)-coordinate of the center. The foci are at \((2, -1 + c)\) and \((2, -1 - c)\).
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Standard Form of an Ellipse
An ellipse can be expressed in the standard form \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \), where \((h, k)\) is the center. The denominators \(a^2\) and \(b^2\) represent the squares of the lengths of the semi-major and semi-minor axes, respectively. Identifying these values helps in graphing the ellipse accurately.
Video consigliato:
5:12
Graph Ellipses at Origin
Determining the Orientation of the Ellipse
The larger denominator between \(a^2\) and \(b^2\) indicates the major axis direction: if \(a^2 > b^2\), the ellipse is stretched horizontally; if \(b^2 > a^2\), it is stretched vertically. This orientation is crucial for locating the foci and sketching the ellipse.
Video consigliato:
4:50Graph Ellipses NOT at Origin
Finding the Foci of an Ellipse
The foci lie along the major axis, located at a distance \(c\) from the center, where \(c = \sqrt{|a^2 - b^2|}\). Knowing \(c\) and the center coordinates allows you to find the exact positions of the foci, which are key points defining the ellipse's shape.
Video consigliato:
5:30Foci and Vertices of an Ellipse
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