Prove that for the general equation of degree 2 Ax² + Bxy + Cy² + Dx + Ey+F = 0 that the angle of rotation 0 need to remove the cross product term in a coordinate transformation is given by A-C cot 20 B
Prove that for the general equation of degree 2 Ax² + Bxy + Cy² + Dx + Ey+F = 0 that the angle of rotation 0 need to remove the cross product term in a coordinate transformation is given by A-C cot 20 B
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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![**Title: Proving the Rotation Angle to Remove the Cross Product Term in a General Equation of Degree 2**
**Objective:** To demonstrate that for the general equation of degree 2:
\[ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \]
the angle of rotation (θ) needed to remove the cross product term (Bxy) in a coordinate transformation is given by the expression:
\[ \cot 2θ = \frac{A - C}{B} \]
**Detailed Explanation:**
To eliminate the cross product term in the quadratic equation, we need to determine the angle θ that will convert the original coordinate system (x, y) to a new coordinate system (x', y') where the cross product term vanishes.
### Step-by-Step Proof:
1. **Original Coordinate System:**
Consider the general quadratic equation:
\[
Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0
\]
2. **New Coordinate System After Rotation:**
When the coordinate system is rotated by an angle θ, the new coordinates (x', y') can be expressed in terms of the original coordinates (x, y) as:
\[
x' = x \cos θ + y \sin θ
\]
\[
y' = -x \sin θ + y \cos θ
\]
3. **Substitute New Coordinates Into the Original Equation:**
Express the original equation in terms of x' and y'. Upon substitution, terms involving \( x'y' \) arise, representing the mixed product term. Our goal is to find θ such that the coefficient of \( x'y' \) becomes zero.
4. **Coefficient of \( x'y' \):**
The coefficient of \( x'y' \) after substitution is computed by focusing on the mixed terms generated by the transformation. This coefficient can be shown as a function of cosine and sine terms, leading to the equation involving cotangent (cot).
5. **Condition for Removing the Cross Product Term:**
For the coefficient of \( x'y' \) to be zero, we obtain the condition:
\[
\cot 2θ = \frac{A - C}{B}
\]
This completes the proof that the angle θ needed to remove the cross product term is given by the above equation. By rotating the coordinate](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F42eb3f21-115e-47f2-b59c-753292f0c3d6%2F2fa4a83a-4de4-4d38-bc9e-a83a6e5c1054%2Fvmgvcyu_processed.png&w=3840&q=75)
Transcribed Image Text:**Title: Proving the Rotation Angle to Remove the Cross Product Term in a General Equation of Degree 2**
**Objective:** To demonstrate that for the general equation of degree 2:
\[ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \]
the angle of rotation (θ) needed to remove the cross product term (Bxy) in a coordinate transformation is given by the expression:
\[ \cot 2θ = \frac{A - C}{B} \]
**Detailed Explanation:**
To eliminate the cross product term in the quadratic equation, we need to determine the angle θ that will convert the original coordinate system (x, y) to a new coordinate system (x', y') where the cross product term vanishes.
### Step-by-Step Proof:
1. **Original Coordinate System:**
Consider the general quadratic equation:
\[
Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0
\]
2. **New Coordinate System After Rotation:**
When the coordinate system is rotated by an angle θ, the new coordinates (x', y') can be expressed in terms of the original coordinates (x, y) as:
\[
x' = x \cos θ + y \sin θ
\]
\[
y' = -x \sin θ + y \cos θ
\]
3. **Substitute New Coordinates Into the Original Equation:**
Express the original equation in terms of x' and y'. Upon substitution, terms involving \( x'y' \) arise, representing the mixed product term. Our goal is to find θ such that the coefficient of \( x'y' \) becomes zero.
4. **Coefficient of \( x'y' \):**
The coefficient of \( x'y' \) after substitution is computed by focusing on the mixed terms generated by the transformation. This coefficient can be shown as a function of cosine and sine terms, leading to the equation involving cotangent (cot).
5. **Condition for Removing the Cross Product Term:**
For the coefficient of \( x'y' \) to be zero, we obtain the condition:
\[
\cot 2θ = \frac{A - C}{B}
\]
This completes the proof that the angle θ needed to remove the cross product term is given by the above equation. By rotating the coordinate
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