Domanda del libro di testo
Find the standard form of the equation of the ellipse satisfying the given conditions. Foci: (-4,0), (4,0); Vertices: (-5,0) (5,0)
2254
views
1
rank
1
comments
Ch. 7 - Conic Sections
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
Non è quello che usi tu?Cambia libro di testoScegli un libro di testo diverso
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 5
Graph each ellipse and locate the foci. x2/25 +y2/64 = 1
Guida verificata passo dopo passo1
Identify the standard form of the ellipse equation given: \(\frac{x^{2}}{25} + \frac{y^{2}}{64} = 1\). Here, \(a^{2}\) and \(b^{2}\) are the denominators under \(x^{2}\) and \(y^{2}\) respectively.
Determine which denominator is larger to find the major axis. Since \(64 > 25\), the major axis is vertical, so \(a^{2} = 64\) and \(b^{2} = 25\).
Calculate the lengths of the semi-major axis \(a\) and semi-minor axis \(b\) by taking the square roots: \(a = \sqrt{64}\) and \(b = \sqrt{25}\).
Find the distance \(c\) from the center to each focus using the relationship \(c^{2} = a^{2} - b^{2}\). Substitute the values of \(a^{2}\) and \(b^{2}\) to find \(c\).
Plot the ellipse centered at the origin with vertices at \((0, \pm a)\) and co-vertices at \((\pm b, 0)\). Then, mark the foci at \((0, \pm c)\) along the major axis.
Risposta video verificata per un problema simile:
Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.
Durata del video:
8mConcetti chiave
Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.
Standard Form of an Ellipse
The equation x²/a² + y²/b² = 1 represents an ellipse centered at the origin. Here, a² and b² are the denominators under x² and y², indicating the lengths of the semi-major and semi-minor axes. Identifying which denominator is larger helps determine the ellipse's orientation.
Video consigliato:
5:12
Graph Ellipses at Origin
Graphing an Ellipse
To graph an ellipse, plot the center at the origin, then mark points a units along the major axis and b units along the minor axis. Connect these points smoothly to form the ellipse. This visual representation helps in understanding the shape and size of the ellipse.
Video consigliato:
4:50Graph Ellipses NOT at Origin
Locating the Foci of an Ellipse
The foci are two fixed points inside the ellipse along the major axis. Their distance from the center is c, found using c² = |a² - b²|. Knowing c allows you to place the foci accurately, which is essential for understanding ellipse properties like reflection and eccentricity.
Video consigliato:
5:30Foci and Vertices of an Ellipse
Pratica correlata
Domanda del libro di testo
Graph each ellipse and locate the foci. x2/9 +y2/36= 1
866
views
Domanda del libro di testo
Find the standard form of the equation of the ellipse satisfying the given conditions. Major axis horizontal with length 12; length of minor axis = 4; center: (-3,5)
1436
views
1
rank
Domanda del libro di testo
Find the standard form of the equation of each hyperbola satisfying the given conditions. Foci: (0, −3), (0, 3) ; vertices: (0, −1), (0, 1)
967
views
Domanda del libro di testo
Find the focus and directrix of the parabola with the given equation. Then graph the parabola.
708
views
Domanda del libro di testo
In Exercises 1–4, find the focus and directrix of each parabola with the given equation. Then match each equation to one of the graphs that are shown and labeled (a)–(d). y^2 = - 4x
918
views