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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 27
Capitolo 8, Problema 27

Find the standard form of the equation of each ellipse satisfying the given conditions. Foci: (0, -4), (0, 4); vertices: (0, −7), (0, 7)

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Identify the center of the ellipse by finding the midpoint of the foci. Since the foci are at (0, -4) and (0, 4), the center is at (0, 0).
Determine the orientation of the ellipse. Because the foci and vertices lie on the y-axis, the major axis is vertical.
Find the distance between the center and each vertex to get the value of \( a \). Here, the vertices are at (0, -7) and (0, 7), so \( a = 7 \).
Find the distance between the center and each focus to get the value of \( c \). The foci are at (0, -4) and (0, 4), so \( c = 4 \).
Use the relationship \( c^2 = a^2 - b^2 \) to solve for \( b^2 \), then write the standard form of the ellipse with a vertical major axis: \$\$ \(\frac{x^2}{b^2}\) + \(\frac{y^2}{a^2}\) = 1 \$\$.

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Standard Form of an Ellipse

The standard form of an ellipse equation depends on the orientation of its major axis. For a vertical major axis centered at the origin, the equation is (x^2 / b^2) + (y^2 / a^2) = 1, where 'a' is the distance from the center to a vertex and 'b' is the distance from the center to a co-vertex.
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Graph Ellipses at Origin

Relationship Between Vertices, Foci, and Axes

In an ellipse, the vertices lie on the major axis at a distance 'a' from the center, while the foci lie on the same axis at a distance 'c'. These distances satisfy the equation c^2 = a^2 - b^2, linking the focal distance, vertex distance, and the minor axis length.
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Foci and Vertices of Hyperbolas

Identifying the Center and Orientation

The center of the ellipse is the midpoint between the vertices and foci. Given the points (0, -4), (0, 4) for foci and (0, -7), (0, 7) for vertices, the center is at the origin (0,0), and the major axis is vertical since all points share the x-coordinate zero.
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Identifying Intervals of Unknown Behavior
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