Skip to main content
Ch. 7 - Conic Sections
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBlitzer 8th EditionCh. 7 - Conic SectionsProblema 25
Capitolo 8, Problema 25

Find the standard form of the equation of each ellipse satisfying the given conditions. Foci: (-5, 0), (5, 0); vertices: (-8, 0), (8,0)

Guida verificata passo dopo passo
1
Identify the center of the ellipse by finding the midpoint of the vertices. Since the vertices are (-8, 0) and (8, 0), the center is at \(\left( \frac{-8 + 8}{2}, \frac{0 + 0}{2} \right) = (0, 0)\).
Determine the orientation of the ellipse. Because the vertices and foci lie on the x-axis, the major axis is horizontal.
Calculate the distance from the center to a vertex, which is the value of \(a\). Here, \(a = 8\).
Calculate the distance from the center to a focus, which is the value of \(c\). Here, \(c = 5\).
Use the relationship \(c^2 = a^2 - b^2\) to find \(b^2\). Rearranged, \(b^2 = a^2 - c^2\). Substitute \(a\) and \(c\) to find \(b^2\), then write the standard form of the ellipse equation as \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\).

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:
2m

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

Standard Form of an Ellipse

The standard form of an ellipse equation depends on the orientation of its major axis. For a horizontal major axis centered at the origin, it is (x²/a²) + (y²/b²) = 1, where 'a' is the distance from the center to a vertex, and 'b' relates to the minor axis length. Understanding this form helps translate geometric information into an algebraic equation.
Video consigliato:
5:12
Graph Ellipses at Origin

Relationship Between Vertices, Foci, and Axes Lengths

In an ellipse, the vertices lie on the major axis at a distance 'a' from the center, while the foci lie at a distance 'c'. These distances satisfy the equation c² = a² - b², linking the focal distance, vertex distance, and minor axis length. Knowing this relationship allows calculation of missing parameters to write the ellipse equation.
Video consigliato:
5:22
Foci and Vertices of Hyperbolas

Center and Orientation of the Ellipse

The center of the ellipse is the midpoint between the vertices and foci. The orientation (horizontal or vertical) is determined by the axis along which the vertices and foci lie. In this problem, both vertices and foci lie on the x-axis, indicating a horizontal major axis centered at the origin, which guides the form of the ellipse equation.
Video consigliato:
4:50
Graph Ellipses NOT at Origin
Pratica correlata
Domanda del libro di testo

Explain why it is not possible for a hyperbola to have foci at (0,-2) and (0,2) and vertices at (0,-3) and (0,3).

1028
views
Domanda del libro di testo

Find the standard form of the equation of the hyperbola satisfying the given conditions. Foci: (-8,0), (8,0); Vertices: (-3,0), (3,0)

1420
views
Domanda del libro di testo

Use vertices and asymptotes to graph each hyperbola. Locate the foci and find the equations of the asymptotes.y=±x2−2y = \(\pm\)\(\sqrt{x^2 - 2}\)

768
views
Domanda del libro di testo

Find the standard form of the equation of each ellipse and give the location of its foci.

848
views
Domanda del libro di testo

Find the standard form of the equation of the hyperbola satisfying the given conditions. Foci: (0,-4), (0,4); Vertices: (0, -2), (0,2)

1010
views
Domanda del libro di testo

Find the standard form of the equation of each parabola satisfying the given conditions. Vertex: (2, - 3); Focus: (2, - 5)

821
views