z² = 15x² +5y²-30
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![### Problem Statement
Reduce the equation to one of the standard forms:
\[ z^2 = 15x^2 + 5y^2 - 30 \]
### Explanation for Students
To reduce the given equation to one of the standard forms, we need to manipulate the equation to identify which standard form it closely represents. Here we have a quadratic equation in three variables \( x \), \( y \), and \( z \).
1. Start by simplifying the equation by combining like terms and factoring if necessary.
2. Compare the simplified form with the standard forms of quadratic surfaces like cones, ellipsoids, hyperboloids, or paraboloids to determine its type.
### Steps to Simplify:
1. **Rewrite the equation**:
\[ z^2 = 15x^2 + 5y^2 - 30 \]
2. **Isolate the constant term**:
\[ z^2 + 30 = 15x^2 + 5y^2 \]
3. **Factor out the coefficients**:
\[ z^2 + 30 = 5(3x^2 + y^2) \]
4. **Divide through by 30 to normalize**:
\[ \frac{z^2}{30} + 1 = \frac{15x^2}{30} + \frac{5y^2}{30} \]
\[ \frac{z^2}{30} + 1 = \frac{x^2}{2} + \frac{y^2}{6} \]
5. **Rearrange to identify the standard form**:
\[ \frac{z^2}{30} = \frac{x^2}{2} + \frac{y^2}{6} - 1 \]
### Conclusion:
The simplified equation can be compared to the standard forms of quadratic surfaces like hyperboloids. Proper manipulation and comparing terms will help you identify the exact form.
Graphs and diagrams are essential to visualizing these equations. Since no graphs or diagrams are provided in the image, ensure to use graphing tools or software to plot the surfaces while studying them.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa2eee30e-7760-4146-894b-912e2799cdb0%2Fbdd84ec7-df9d-4df7-bfc5-248914abca2c%2Fq37v30q_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Statement
Reduce the equation to one of the standard forms:
\[ z^2 = 15x^2 + 5y^2 - 30 \]
### Explanation for Students
To reduce the given equation to one of the standard forms, we need to manipulate the equation to identify which standard form it closely represents. Here we have a quadratic equation in three variables \( x \), \( y \), and \( z \).
1. Start by simplifying the equation by combining like terms and factoring if necessary.
2. Compare the simplified form with the standard forms of quadratic surfaces like cones, ellipsoids, hyperboloids, or paraboloids to determine its type.
### Steps to Simplify:
1. **Rewrite the equation**:
\[ z^2 = 15x^2 + 5y^2 - 30 \]
2. **Isolate the constant term**:
\[ z^2 + 30 = 15x^2 + 5y^2 \]
3. **Factor out the coefficients**:
\[ z^2 + 30 = 5(3x^2 + y^2) \]
4. **Divide through by 30 to normalize**:
\[ \frac{z^2}{30} + 1 = \frac{15x^2}{30} + \frac{5y^2}{30} \]
\[ \frac{z^2}{30} + 1 = \frac{x^2}{2} + \frac{y^2}{6} \]
5. **Rearrange to identify the standard form**:
\[ \frac{z^2}{30} = \frac{x^2}{2} + \frac{y^2}{6} - 1 \]
### Conclusion:
The simplified equation can be compared to the standard forms of quadratic surfaces like hyperboloids. Proper manipulation and comparing terms will help you identify the exact form.
Graphs and diagrams are essential to visualizing these equations. Since no graphs or diagrams are provided in the image, ensure to use graphing tools or software to plot the surfaces while studying them.
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