(x-4)2, (y+2)2 10 2. = 1 25 9. 8- Center: Vertices: -10 10 Foci: -8 10-
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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Question
Part 1 - Ellipse: Graph each ellipse and locate the foci. For full credit, you must show all your work!
Graph each ellipse and locate the foci. For full credit, you must show all your work!
![## Equation of the Ellipse
**Equation:**
\[ \frac{(x-4)^2}{9} + \frac{(y+2)^2}{25} = 1 \]
**Center:**
To find the center of the ellipse, identify the values that shift the origin, \((h, k)\):
- \(h = 4\)
- \(k = -2\)
The center is at \((4, -2)\).
**Vertices:**
For an ellipse in the form \(\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\), the larger denominator corresponds to the major axis.
- Major Axis along the \(y\)-axis: \(b^2 = 25 \Rightarrow b = 5\)
- Minor Axis along the \(x\)-axis: \(a^2 = 9 \Rightarrow a = 3\)
Vertices are at \((4, -2 \pm 5)\), i.e., \((4, 3)\) and \((4, -7)\).
**Foci:**
The distance from the center to each focus, \(c\), is found using \(c^2 = b^2 - a^2\).
- \(c^2 = 25 - 9 = 16 \Rightarrow c = 4\)
Foci are at \((4, -2 \pm 4)\), i.e., \((4, 2)\) and \((4, -6)\).
## Graph Explanation
The graph is a standard coordinate plane with grids progressing from \(-10\) to \(10\) on both axes. The graph aids in plotting the ellipse centered at \((4, -2)\) based on the given equation. It shows:
- Horizontal span defined by \(\pm 3\) from the center.
- Vertical span defined by \(\pm 5\) from the center.
This provides a visual guide for sketching and understanding the properties of the ellipse, including its orientation, shape, and important points like vertices and foci.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F16ae3d29-122f-4fea-a088-af0cdad091d7%2Ffbacc434-aa84-4dce-93fd-afe3b7ac531f%2Fyo44ybb_processed.jpeg&w=3840&q=75)
Transcribed Image Text:## Equation of the Ellipse
**Equation:**
\[ \frac{(x-4)^2}{9} + \frac{(y+2)^2}{25} = 1 \]
**Center:**
To find the center of the ellipse, identify the values that shift the origin, \((h, k)\):
- \(h = 4\)
- \(k = -2\)
The center is at \((4, -2)\).
**Vertices:**
For an ellipse in the form \(\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\), the larger denominator corresponds to the major axis.
- Major Axis along the \(y\)-axis: \(b^2 = 25 \Rightarrow b = 5\)
- Minor Axis along the \(x\)-axis: \(a^2 = 9 \Rightarrow a = 3\)
Vertices are at \((4, -2 \pm 5)\), i.e., \((4, 3)\) and \((4, -7)\).
**Foci:**
The distance from the center to each focus, \(c\), is found using \(c^2 = b^2 - a^2\).
- \(c^2 = 25 - 9 = 16 \Rightarrow c = 4\)
Foci are at \((4, -2 \pm 4)\), i.e., \((4, 2)\) and \((4, -6)\).
## Graph Explanation
The graph is a standard coordinate plane with grids progressing from \(-10\) to \(10\) on both axes. The graph aids in plotting the ellipse centered at \((4, -2)\) based on the given equation. It shows:
- Horizontal span defined by \(\pm 3\) from the center.
- Vertical span defined by \(\pm 5\) from the center.
This provides a visual guide for sketching and understanding the properties of the ellipse, including its orientation, shape, and important points like vertices and foci.
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