Problem 12 - Precalculus: Mathematical for Calculus

Question

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 conic equation in rotated coordinates is \(X^2 - 2\sqrt{2}X - Y^2 = 0\).

Step 1: Identify Original Conic

The given conic equation is \(xy = x + y\). This is a non-standard form of a rectangular hyperbola.

Step 2: Determine Rotation Matrix

The conic is to be rotated by an angle \(\phi = \frac{\pi}{4}\). The rotation matrix for an angle \(\phi\) is given by:\[R = \begin{bmatrix} \cos\phi & -\sin\phi \ \sin\phi & \cos\phi \end{bmatrix}\] Substitute \(\phi = \frac{\pi}{4}\):\[R = \begin{bmatrix} \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \ \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{bmatrix}\]

Step 3: Perform Coordinate Transformation

To transform the coordinates, we use the substitution:\[x = X\cos\phi - Y\sin\phi, \quad y = X\sin\phi + Y\cos\phi\]Plug in \(\phi = \frac{\pi}{4}\):\[x = \frac{X}{\sqrt{2}} - \frac{Y}{\sqrt{2}}, \quad y = \frac{X}{\sqrt{2}} + \frac{Y}{\sqrt{2}}\]

Step 4: Substitute Transformed Coordinates into Original Equation

Substitute the expressions for \(x\) and \(y\) in terms of \(X\) and \(Y\) into the original equation \(xy = x + y\):\[\left(\frac{X}{\sqrt{2}} - \frac{Y}{\sqrt{2}}\right) \left(\frac{X}{\sqrt{2}} + \frac{Y}{\sqrt{2}}\right) = \frac{X}{\sqrt{2}} - \frac{Y}{\sqrt{2}} + \frac{X}{\sqrt{2}} + \frac{Y}{\sqrt{2}}\]Simplifying both sides gives:\[\frac{X^2}{2} - \frac{Y^2}{2} = \sqrt{2}X\]

Step 5: Rearrange into Conic Equation

Multiply the entire equation by 2 to eliminate the fractions:\[ X^2 - Y^2 = 2\sqrt{2}X \]Therefore, the equation of the conic in new \(XY\)-coordinates is:\[ X^2 - 2\sqrt{2}X - Y^2 = 0 \]

Key Concepts

Coordinate Transformation

Understanding how to transform coordinates is essential in geometry and trigonometry, especially when dealing with conic sections. When a conic section, like a hyperbola, undergoes a rotation, its coordinates need to be transformed to reflect this change. This allows for analyzing the conic in a more convenient axis orientation. When performing a coordinate transformation, new coordinates are calculated using existing ones, considering a certain angle of rotation.
One common method of coordinate transformation involves using trigonometric functions such as sine and cosine. By rotating the axes, you can express the new coordinates (often noted as \(X\) and \(Y\)) in terms of the original coordinates (\(x\) and \(y\)). The transformation is given by the equations:

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

Where \(\phi\) is the angle through which the axes are rotated. In the given problem, a transformation is made using \(\phi = \pi/4\) radians, which simplifies the analysis of the hyperbola.

Rotation Matrix

The rotation matrix is a fundamental concept used in the transformation of coordinates. It provides a structured way to rotate points in the coordinate plane.
A rotation matrix for an angle \(\phi\) is a 2x2 matrix represented as:
\[ R = \begin{bmatrix} \cos(\phi) & -\sin(\phi) \ \sin(\phi) & \cos(\phi) \end{bmatrix} \]This matrix, when applied, rotates a point or a conic section through the specified angle \(\phi\).
To find the new coordinates of a conic section after rotation, the original coordinates are multiplied by this rotation matrix. With \(\phi = \pi/4\), the matrix becomes:
\[ R = \begin{bmatrix} \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \ \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{bmatrix} \]These values are derived from the trigonometric values at \(45^\circ\) or \(\pi/4\) radians. This rotation matrix is then used to transform each point of the conic section to its new orientation, making the equation easier to work with.

Rectangular Hyperbola

Rectangular hyperbolas are a special kind of hyperbola where the asymptotes are perpendicular to one another, usually at 90 degrees. This distinctive feature allows the hyperbola to possess symmetrical properties, making it relatively easier to analyze.
The general form of a rectangular hyperbola in terms of \(x\) and \(y\) coordinates is \(xy = c\), where \(c\) is some constant. In this exercise, the original equation \(xy = x + y\) represents such a hyperbola.
After applying the rotation transformation indicated by \(\phi = \pi/4\), the hyperbola's equation transforms into another form. This outcome is the rotated equation that still maintains the geometric nature of a rectangular hyperbola but in a new coordinate system \((X, Y)\).
Such transformations and rotations are invaluable in simplifying complex conics for easier interpretation and solution. They allow for an exploration of different properties and insights that might not be apparent in the original orientation. In this specific solution, the final equation becomes \(X^2 - 2\sqrt{2}X - Y^2 = 0\), clearly indicating the transformation effects of the rectangular hyperbola.