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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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

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.

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.