Problem 42 - Thomas Calculus

Question

Show that rotating the axes through an angle of \(\pi / 4\) radians will eliminate the \(x y\) -term from Equation \((1)\) whenever \(A=C\) .

Step-by-Step Solution

Answer

The \( xy \)-term is eliminated by rotating the axes through \( \frac{\pi}{4} \) radians when \( A = C \).

Step 1: Understand the Original Equation

The original equation of a conic section is typically written in the form:\[ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \]For this problem, we know that \( A = C \), which means the equation simplifies to:\[ Ax^2 + Bxy + Ax^2 + Dx + Ey + F = 0 \]Our goal is to eliminate the \( Bxy \) term by rotating the axes.

Step 2: Coordinate Transformation

We will perform a rotation of the axes by an angle \( \theta \), where \( \theta = \frac{\pi}{4} \) radians. The transformation equations for the rotation are:\[ x = X\cos(\theta) - Y\sin(\theta) \]\[ y = X\sin(\theta) + Y\cos(\theta) \]Substituting these into the original equation will transform the coordinates from \((x, y)\) to \((X, Y)\).

Step 3: Substitute and Simplify

Substitute \( x = X\cos\left(\frac{\pi}{4}\right) - Y\sin\left(\frac{\pi}{4}\right) \) and \( y = X\sin\left(\frac{\pi}{4}\right) + Y\cos\left(\frac{\pi}{4}\right) \) into the equation. Since both \( \sin\left(\frac{\pi}{4}\right) \) and \( \cos\left(\frac{\pi}{4}\right) \) are \( \frac{\sqrt{2}}{2} \), the transformations become:\[ x = \frac{\sqrt{2}}{2}(X - Y) \]\[ y = \frac{\sqrt{2}}{2}(X + Y) \]Substitute these into the original equation.

Step 4: Calculate the New Expression for \(Bxy\)

The original \( Bxy \) term transforms as follows:\[ Bxy = B\left(\frac{\sqrt{2}}{2}(X - Y)\right)\left(\frac{\sqrt{2}}{2}(X + Y)\right) \]Simplify this product to:\[ Bxy = \frac{B}{2}((X^2 - Y^2)) \]Simplifying further leads to:\[ Bxy = \frac{B}{2}(X^2 - Y^2) \]

Step 5: Eliminate the \(Bxy\) Term

After substitution and simplification, the complete equation in \((X, Y)\) coordinates has the transformed expression of the \(Bxy\) term and by substituting \( A = C \), the overall transformation makes the coefficient of the \(XY\) term vanish due to the angle \( \frac{\pi}{4} \) being exactly offset such that:\[ B = 2(A-C) \sin(2\theta) \]Thus the \( XY \) term eliminates when \( A = C \).

Step 6: Conclusion

In summary, rotating the axes by \( \theta = \frac{\pi}{4} \) radians with the condition \( A = C \) causes the mixed \( XY \) term to vanish in the new coordinate system. The transformation effectively aligns the axes with the direction of the conic axes, thus simplifying the equation.

Key Concepts

Rotation of Axes

In geometry, when we face an equation with both \(x^2\) and \(xy\) terms, often a rotation of the axes can help simplify it. This is particularly useful for conic sections, which include ellipses, hyperbolas, and parabolas. Rotating the axes means we are changing our perspective without altering the shape of the graph, just like tilting your head but the picture remains the same.

To rotate the axes by an angle \( \theta \), we transform the original \((x, y)\) coordinates to new coordinates, \((X, Y)\), using:

• \(x = X \cos(\theta) - Y \sin(\theta)\)
• \(y = X \sin(\theta) + Y \cos(\theta)\)

This rotation can strategically eliminate the messy \(xy\) terms, making the math neater! For the exercise, rotating by \( \theta = \frac{\pi}{4} \) radians simplifies the task, perfectly aligning our axes with the natural orientation of the shape.

Coordinate Transformation

Performing a coordinate transformation is like putting on a new pair of glasses; it gives us a fresh view of the equation. Particularly, when we rotate the axes by \( \theta = \frac{\pi}{4} \) radians:

• The values \(\cos(\frac{\pi}{4})\) and \(\sin(\frac{\pi}{4})\) are both \(\frac{\sqrt{2}}{2}\).
• Through substitution, \(x = \frac{\sqrt{2}}{2}(X - Y)\)
• And \(y = \frac{\sqrt{2}}{2}(X + Y)\)

These transformations convert the original \((x, y)\) coordinates into \((X, Y)\), providing a new lens to explore the equation. By mapping old coordinates onto new ones, we can focus on the true orientation and shape of the conic, simplifying calculations and reducing complexity.

Mixed xy Term Elimination

In some conic sections, we encounter an \(xy\) term that makes the equation cumbersome. But this ugly term can be eliminated by a wise choice of rotation, making graph interpretation child's play. The exercise focuses precisely on eliminating the \(Bxy\) part when \(A = C\):

• Substituting into the transformed coordinates, \(Bxy = \frac{B}{2}(X^2 - Y^2)\).
• The \(XY\) term disappears by making sure \(B = 2(A-C) \sin(2\theta)\).
• By aligning \(\theta\) to \(\frac{\pi}{4}\) and with \(A = C\), \(XY\) naturally vanishes, resulting in no mixed term left.

Through this transformation and rotation, the original equation is stripped of its complexity. It highlights the ability of geometric rotations to simplify equations and offer insights into their structure.