Problem 14 - Algebra and Trigonometry

Question

\(9-14\) . Determine the equation of the given conic in \(X Y\) -coordinates when the coordinate axes are rotated through the indicated angle. $$ x y=x+y, \quad \phi=\pi / 4 $$

Step-by-Step Solution

Answer

The rotated conic's equation is \( X^2 - Y^2 = 2\sqrt{2}X \).

Step 1: Identify the Problem Type

The given problem involves finding the new equation of a conic after rotating the coordinate axes. The rotation angle is given as \( \phi = \frac{\pi}{4} \).

Step 2: Recall Rotation Transformation Formulas

The formulas to transform coordinates through an angle \( \phi \) are: \[ x = X \cos\phi - Y \sin\phi \] \[ y = X \sin\phi + Y \cos\phi \] These will be used to convert the original equation \( xy = x + y \) into the new coordinate system.

Step 3: Substitute Rotation Formulas

Substitute the transformation equations into the original equation \( xy = x + y \) with \( \phi = \frac{\pi}{4} \): \[ (X \cos\phi - Y \sin\phi)(X \sin\phi + Y \cos\phi) = X \cos\phi - Y \sin\phi + X \sin\phi + Y \cos\phi. \]

Step 4: Simplify Trigonometric Values

Since \( \cos\frac{\pi}{4} = \sin\frac{\pi}{4} = \frac{\sqrt{2}}{2} \), update the equation:\[ \left(X \frac{\sqrt{2}}{2} - Y \frac{\sqrt{2}}{2}\right) \left(X \frac{\sqrt{2}}{2} + Y \frac{\sqrt{2}}{2}\right) = X \frac{\sqrt{2}}{2} - Y \frac{\sqrt{2}}{2} + X \frac{\sqrt{2}}{2} + Y \frac{\sqrt{2}}{2}. \]

Step 5: Expand and Combine Terms

Expand the expression:\[ \frac{1}{2}(X^2 - Y^2) = \frac{\sqrt{2}}{2}(2X) \Rightarrow X^2 - Y^2 = 2\sqrt{2}X. \]

Step 6: Final Answer

The equation of the conic in the new rotated \( XY \) coordinates is: \[ X^2 - Y^2 = 2\sqrt{2}X. \]

Key Concepts

Coordinate Rotation

When tackling the problem of transforming a conic section, one major step is the rotation of coordinates. Coordinate rotation is essentially adjusting the axes so that the conic section aligns in a way that simplifies the problem.
For instance, when a conic section is rotated through a given angle, such as \( \phi = \frac{\pi}{4} \), it means we are turning our view of the plane so that everything shifts according to this angle. This can help us to see some underlying symmetry or simplification in the conic equation.

• The process involves mathematical transformation using trigonometric relationships.
• This perspective change is fundamentally about reorienting our coordinate axes.

This step is particularly important as the new alignment of the axes can lead to simplifying the conic equations, making it easier to understand and solve.

Transformation Formulas

To successfully rotate the coordinate axes, transformation formulas are crucial. These formulas translate old coordinates \((x, y)\) into new coordinates \((X, Y)\) after rotation.
The key transformation formulas used are:

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

In our example with \(\phi = \frac{\pi}{4}\), these formulas help to transition from our original base equation into the rotated coordinate system.
Here, the role of the trigonometric values of \( \cos \) and \( \sin \) seen within these formulas is vital, as they dictate how the coordinates warp and compress during rotation. Mastery of this translation through practice is critical for tackling complex geometric transformations.

Trigonometric Values

Trigonometric values are fundamental in solving and simplifying conic section equations, especially when coordinate rotation is involved. For \( \phi = \frac{\pi}{4} \), both \( \cos\) and \(\sin\) equal \( \frac{\sqrt{2}}{2} \).

• This unique property at \( \frac{\pi}{4} \) often leads to clean, symmetrical simplifications.
• Understanding these values allows us to correctly substitute into transformation formulas.

Therefore, the more you familiarize yourself with common trigonometric values and their respective angles, the more effectively you can apply transformations and solve for new, simpler conic equations.
Making these connections between angles and their trigonometric functions helps enhance your skill in both geometry and algebra.

Equation Simplification

Simplifying equations post-rotation is the final, crucial step in understanding conic sections through coordinate transformations. Here, the goal is to rework the complex, transformed equation into something simpler.
In our solved example, after transformation, the equation simplifies to \( X^2 - Y^2 = 2\sqrt{2}X \).
The steps involve:

• Expanding any multiplied terms (like polynomials).
• Combining like terms to consolidate the equation.
• Eventually revealing an equation that, hopefully, shows some pattern or symmetry.

These processes align with core algebra skills and principles, allowing one to distill the problem to its purest mathematical form.
Learning how to simplify these equations after applying transformations is essential in solving conic section problems efficiently and accurately.