Given the ellipse 25 y? =D1 16 List the coordinates of the foci.
Trigonometry (11th Edition)
11th Edition
ISBN:9780134217437
Author:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Chapter1: Trigonometric Functions
Section: Chapter Questions
Problem 1RE:
1. Give the measures of the complement and the supplement of an angle measuring 35°.
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![**Problem Statement:**
Given the ellipse
\[
\frac{x^2}{25} + \frac{y^2}{16} = 1
\]
List the coordinates of the foci.
**Explanation:**
This is a standard equation of an ellipse centered at the origin. The terms \(\frac{x^2}{a^2}\) and \(\frac{y^2}{b^2}\) determine the semi-major and semi-minor axes.
- Here, \(a^2 = 25\) and \(b^2 = 16\).
- Thus, \(a = 5\) and \(b = 4\).
Since \(a > b\), the major axis is along the x-axis.
The distance of the foci from the center (c) can be found using the formula:
\[ c = \sqrt{a^2 - b^2} \]
\[ c = \sqrt{25 - 16} \]
\[ c = \sqrt{9} \]
\[ c = 3 \]
Therefore, the coordinates of the foci are \((\pm 3, 0)\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9a5a9e76-fe13-4ee9-bb36-e44a23423bab%2Fbc994a28-2bb2-4a97-a400-a42b44d85ca6%2Fvtbnk4b_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Given the ellipse
\[
\frac{x^2}{25} + \frac{y^2}{16} = 1
\]
List the coordinates of the foci.
**Explanation:**
This is a standard equation of an ellipse centered at the origin. The terms \(\frac{x^2}{a^2}\) and \(\frac{y^2}{b^2}\) determine the semi-major and semi-minor axes.
- Here, \(a^2 = 25\) and \(b^2 = 16\).
- Thus, \(a = 5\) and \(b = 4\).
Since \(a > b\), the major axis is along the x-axis.
The distance of the foci from the center (c) can be found using the formula:
\[ c = \sqrt{a^2 - b^2} \]
\[ c = \sqrt{25 - 16} \]
\[ c = \sqrt{9} \]
\[ c = 3 \]
Therefore, the coordinates of the foci are \((\pm 3, 0)\).
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