Q: Discuss find the following surfaces and Traces. intercepts and 4x² + +4y.2 + 2²-16=0
Q: Discuss find the following surfaces and Traces. intercepts and 4x² + +4y.2 + 2²-16=0
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter11: Topics From Analytic Geometry
Section11.2: Ellipses
Problem 33E
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![### Surface Analysis: Intercepts and Traces Discussion
#### Problem Statement
**Question:**
Discuss the following surface and find intercepts and traces.
\[ 4x^2 + 4y^2 + z^2 - 16 = 0. \]
#### Explanation and Steps
1. **Identify the Surface:**
The given equation is a standard form that represents a three-dimensional surface. Upon rearranging the equation to a more familiar form, you get:
\[ 4x^2 + 4y^2 + z^2 = 16. \]
2. **Simplify the Equation:**
Divide each term by 16:
\[ \frac{4x^2}{16} + \frac{4y^2}{16} + \frac{z^2}{16} = 1, \]
which simplifies to:
\[ \frac{x^2}{4} + \frac{y^2}{4} + \frac{z^2}{16} = 1. \]
3. **Surface Type:**
This simplified form represents an ellipsoid, which is symmetric with respect to the x, y, and z axes.
4. **Intercepts:**
- **x-intercepts:**
Set \( y = 0 \) and \( z = 0 \):
\[ \frac{x^2}{4} = 1 \Rightarrow x^2 = 4 \Rightarrow x = \pm 2. \]
Therefore, the x-intercepts are \( (2, 0, 0) \) and \( (-2, 0, 0) \).
- **y-intercepts:**
Set \( x = 0 \) and \( z = 0 \):
\[ \frac{y^2}{4} = 1 \Rightarrow y^2 = 4 \Rightarrow y = \pm 2. \]
Therefore, the y-intercepts are \( (0, 2, 0) \) and \( (0, -2, 0) \).
- **z-intercepts:**
Set \( x = 0 \) and \( y = 0 \):
\[ \frac{z^2}{16} = 1 \Rightarrow z^2 = 16 \Rightarrow z = \](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F260936ea-709c-4c58-af77-5b65d085fea3%2F89a65c10-04fe-47ed-ae6b-81a88494c490%2F51c2j1q_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Surface Analysis: Intercepts and Traces Discussion
#### Problem Statement
**Question:**
Discuss the following surface and find intercepts and traces.
\[ 4x^2 + 4y^2 + z^2 - 16 = 0. \]
#### Explanation and Steps
1. **Identify the Surface:**
The given equation is a standard form that represents a three-dimensional surface. Upon rearranging the equation to a more familiar form, you get:
\[ 4x^2 + 4y^2 + z^2 = 16. \]
2. **Simplify the Equation:**
Divide each term by 16:
\[ \frac{4x^2}{16} + \frac{4y^2}{16} + \frac{z^2}{16} = 1, \]
which simplifies to:
\[ \frac{x^2}{4} + \frac{y^2}{4} + \frac{z^2}{16} = 1. \]
3. **Surface Type:**
This simplified form represents an ellipsoid, which is symmetric with respect to the x, y, and z axes.
4. **Intercepts:**
- **x-intercepts:**
Set \( y = 0 \) and \( z = 0 \):
\[ \frac{x^2}{4} = 1 \Rightarrow x^2 = 4 \Rightarrow x = \pm 2. \]
Therefore, the x-intercepts are \( (2, 0, 0) \) and \( (-2, 0, 0) \).
- **y-intercepts:**
Set \( x = 0 \) and \( z = 0 \):
\[ \frac{y^2}{4} = 1 \Rightarrow y^2 = 4 \Rightarrow y = \pm 2. \]
Therefore, the y-intercepts are \( (0, 2, 0) \) and \( (0, -2, 0) \).
- **z-intercepts:**
Set \( x = 0 \) and \( y = 0 \):
\[ \frac{z^2}{16} = 1 \Rightarrow z^2 = 16 \Rightarrow z = \
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