**Educational Mathematics Content: Rotating Axes to Eliminate the xy-Term**---### Problem StatementPerform 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 GraphsThe 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.---

Given: 9x2-63xy+3y2+6x+63y=0 1 Putting x=x'cosθ-y'sinθ , y=x'sinθ+y'cosθ in (1), we get…

Answered: rm a rotation of axes to eliminate the…

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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Concept explainers

Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.

Question

**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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