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Coordinate transformation is a fundamental process in geometry that lets us change the perspective of a problem by altering the coordinates. Imagine it like spinning a map to see it from a different angle. When working with an equation involving both \(x\) and \(y\), transforming coordinates can make the equation simpler to solve. For this purpose, mathematical rotations are used.
In the case of rotating axes, we use trigonometric functions. For any point originally at \( (x, y) \), we calculate new coordinates \( (x', y') \) using the rotation formulas:

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

By choosing an appropriate angle, such as \(45°\) in this problem, you can remove cross-product terms like \(xy\) that complicate solving the equation. As a result, the new equation can often be far easier to interpret and solve.

The standard form of any conic section, such as ellipses, hyperbolas, or parabolas, typically has a simpler expression without mixed terms like \(xy\). After the transformation, the goal is to rearrange the terms to place the equation in its standard form.
In the exercise context, the given equation \(xy - 8x - 4y = 0\) is rotated to facilitate dropping the \(xy\) term. The new coordinate expressions for \( x' \) and \( y' \) allow you to streamline transformations and achieve the standard form:
\[ x' = \frac{4y'}{y-8} \]
This equation can now be a lot simpler to evaluate or graph, compared to its original incarnation. The standard form provides a more efficient pathway to analyze and interpret conic sections.

The angle of rotation is a key element when modifying axes. It determines how much we "turn" the coordinate system to make the equation more straightforward. For equations containing mixed terms like \(xy\), a common choice for angle of rotation is \(45°\) or \(\pi/4\) radians.
By choosing this angle, the formulas:

• \( x = (x+y)/\sqrt{2} \)
• \( y = (y-x)/\sqrt{2} \)

will rotate the axes effectively, removing the troublesome \(xy\) term. The reason \(45°\) is often selected is its mathematical properties in removing these terms directly due to the sine and cosine values being equal at this angle, i.e., \(\sin(45°) = \cos(45°) = 1/\sqrt{2}\).
Understanding why and how we choose angles lets us strategically rotate axes, optimizing the process of finding simpler, more insightful mathematical solutions.