Given the equation of an ellipse:\[ \frac{x^2}{16} + \frac{y^2}{25} = 1 \]Determine the endpoints of the minor and major axes for the graph of this ellipse.- **Maximum point on the major axis:** [ ]- **Minimum point on the major axis:** [ ]- **Maximum point on the minor axis:** [ ]- **Minimum point on the minor axis:** [ ]- **Maximum focal point:** [ ]- **Minimum focal point:** [ ]**Explanation:**- The ellipse is centered at the origin (0,0).- The major axis is vertical since 25 > 16.- The lengths of the semi-major axis (a) and semi-minor axis (b) are derived from the denominators under the x and y terms, where a² = 25 and b² = 16.- The major axis length is 2a = 10, and the minor axis length is 2b = 8.- Foci can be found using $ c = \sqrt{a^2 - b^2} $.Fill in the calculated values to find specific points and foci.

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y Let 16 25 then find the end points of the minor and major axis for the graph of this ellipse. Maximum point on the major axis: Minimum point on the major axis: Maximum point on the minor axis: Minimum point on the minor axis: Maximum focal point: Minimum focal point:

y Let 16 25 then find the end points of the minor and major axis for the graph of this ellipse. Maximum point on the major axis: Minimum point on the major axis: Maximum point on the minor axis: Minimum point on the minor axis: Maximum focal point: Minimum focal point:

Algebra and Trigonometry (6th Edition)

6th Edition

ISBN:9780134463216

Author:Robert F. Blitzer

Publisher:Robert F. Blitzer

ChapterP: Prerequisites: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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Concept explainers

Vertices Of An Ellipse

In the study of the conic section of the geometry, An ellipse is a plane curve surrounding two focal points, or foci where the set of all locus of points in the plane has their sum of the distances to the two foci is constant. On the ellipse curve, it is the intersection point of the main axis. In an ellipse graph, the center of the axis is the midpoint of the line segment connecting the foci. The major axis is the line section that runs through the ellipse's foci, while the minor axis runs through the middle and is perpendicular to the major axis.. The ellipse has two endpoints on the major axis which are known as the vertices (in plural), vertex in the singular. The equation of the major axis is 2a, the equation of the minor is 2b and the distance between the major axis and minor axis is 2c. The semi-major axis has a length of a and the semi-minor axis has a length of b. The ellipse's vertices are determined by the elliptical orbit's equation and graph.

Question

Given the equation of an ellipse:

\[ \frac{x^2}{16} + \frac{y^2}{25} = 1 \]

Determine the endpoints of the minor and major axes for the graph of this ellipse.

- **Maximum point on the major axis:** [ ]
- **Minimum point on the major axis:** [ ]

- **Maximum point on the minor axis:** [ ]
- **Minimum point on the minor axis:** [ ]

- **Maximum focal point:** [ ]
- **Minimum focal point:** [ ]

**Explanation:**
- The ellipse is centered at the origin (0,0).
- The major axis is vertical since 25 > 16.
- The lengths of the semi-major axis (a) and semi-minor axis (b) are derived from the denominators under the x and y terms, where a² = 25 and b² = 16.
- The major axis length is 2a = 10, and the minor axis length is 2b = 8.
- Foci can be found using $ c = \sqrt{a^2 - b^2} $.

Fill in the calculated values to find specific points and foci.

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