rm a rotation of axes to eliminate the xy-term. (Use x2 and y2 for the rotated coordinates.) 22 - 6/3xy + 3y? + 6x + 6V3y = 0 ch the graph of the conic.
rm a rotation of axes to eliminate the xy-term. (Use x2 and y2 for the rotated coordinates.) 22 - 6/3xy + 3y? + 6x + 6V3y = 0 ch the graph of the conic.
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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![**Educational Mathematics Content: Rotating Axes to Eliminate the xy-Term**
---
### Problem Statement
Perform a rotation of axes to eliminate the xy-term in the given quadratic equation. Use \( x_2 \) and \( y_2 \) for the rotated coordinates.
Given Equation:
\[ 9x^2 - 6\sqrt{3}xy + 3y^2 + 6x + 6\sqrt{3}y = 0 \]
---
### Task
**Sketch the graph of the conic.**
---
### Explanation of Graphs
The image displays four graphs depicting the same conic section, oriented differently to demonstrate various steps or perspectives of the transformation.
1. **First Graph (Top Left):**
- Displays a rotated conic section, likely illustrating the original equation prior to any transformation.
- The axes are labeled \( x \) and \( y \), extending from -4 to 4 on both axes.
2. **Second Graph (Top Right):**
- Another view of the conic section after an initial step of manipulation.
- Similar scale and axis labeling are maintained.
3. **Third Graph (Bottom Left):**
- Here, a further transformation may have been applied, suggesting the progression in eliminating the xy-term.
- The scale remains consistent with previous graphs, aiding in comparing transformations.
4. **Fourth Graph (Bottom Right):**
- This likely represents the final configuration of the conic after successful rotation to eliminate the xy-term.
- The graph displays the conic more aligned with the axes.
Each graph uses the same axes limits to provide a consistent view, ensuring that transformations can be compared visually. The equations of the transformations are suggested in the problem statement, with a focus on rotating to clarify the conic's form.
These diagrams visually illustrate the process of axis rotation, a crucial technique in conic section analysis, and provide insights into the effects on graphical representation concerning rotation in coordinate geometry.
---](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F350a872a-0bb3-4218-bc06-9a68f5ae92af%2F7f039f24-5a71-465f-92b8-4cf6d1aa831a%2Fnre123v_processed.png&w=3840&q=75)
Transcribed Image Text:**Educational Mathematics Content: Rotating Axes to Eliminate the xy-Term**
---
### Problem Statement
Perform a rotation of axes to eliminate the xy-term in the given quadratic equation. Use \( x_2 \) and \( y_2 \) for the rotated coordinates.
Given Equation:
\[ 9x^2 - 6\sqrt{3}xy + 3y^2 + 6x + 6\sqrt{3}y = 0 \]
---
### Task
**Sketch the graph of the conic.**
---
### Explanation of Graphs
The image displays four graphs depicting the same conic section, oriented differently to demonstrate various steps or perspectives of the transformation.
1. **First Graph (Top Left):**
- Displays a rotated conic section, likely illustrating the original equation prior to any transformation.
- The axes are labeled \( x \) and \( y \), extending from -4 to 4 on both axes.
2. **Second Graph (Top Right):**
- Another view of the conic section after an initial step of manipulation.
- Similar scale and axis labeling are maintained.
3. **Third Graph (Bottom Left):**
- Here, a further transformation may have been applied, suggesting the progression in eliminating the xy-term.
- The scale remains consistent with previous graphs, aiding in comparing transformations.
4. **Fourth Graph (Bottom Right):**
- This likely represents the final configuration of the conic after successful rotation to eliminate the xy-term.
- The graph displays the conic more aligned with the axes.
Each graph uses the same axes limits to provide a consistent view, ensuring that transformations can be compared visually. The equations of the transformations are suggested in the problem statement, with a focus on rotating to clarify the conic's form.
These diagrams visually illustrate the process of axis rotation, a crucial technique in conic section analysis, and provide insights into the effects on graphical representation concerning rotation in coordinate geometry.
---
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