Problem 47 - Calculus Early Transcendentals: Pearson New International Edition

Question

Eliminate the cross-product term by a suitable rotation of axes and then, if necessary, translate axes (complete the squares) to put the equation in standard form. Finally, graph the equation showing the rotated axes. $$ -\frac{1}{2} x^{2}+7 x y-\frac{1}{2} y^{2}-6 \sqrt{2} x-6 \sqrt{2} y=0 $$

Step-by-Step Solution

Answer

Rotate by \(45^\circ\), equation is a parabola: \(-Y^2 = -3\sqrt{2}X\).

Step 1: Identify the cross-product term

The given quadratic equation is \(-\frac{1}{2} x^{2}+7 x y-\frac{1}{2} y^{2}-6 \sqrt{2} x-6 \sqrt{2} y=0\). We notice the cross-product term is \(7xy\). Our goal is to eliminate this term by rotating the axes.

Step 2: Determine the angle of rotation

For an equation of the form \(Ax^2 + Bxy + Cy^2\), the angle \(\theta\) for rotation is found using \(\tan(2\theta) = \frac{B}{A-C}\). For our equation, \(A = -\frac{1}{2}\), \(B = 7\), \(C = -\frac{1}{2}\). So, \(\tan(2\theta) = \frac{7}{0} = \text{undefined}\). Thus, \(2\theta = \frac{\pi}{2}\) and \(\theta = \frac{\pi}{4}\).

Step 3: Apply the rotation of axes

Define new variables: \(x = X\cos(\theta) - Y\sin(\theta)\) and \(y = X\sin(\theta) + Y\cos(\theta)\), where \(\theta = \frac{\pi}{4}\). Substitute and simplify the equation: The new variables \(X, Y\) will eliminate the \(XY\) term. Substitute \(x = \frac{X-Y}{\sqrt{2}}\) and \(y = \frac{X+Y}{\sqrt{2}}\) into the equation.

Step 4: Substitute expressions and simplify

Substitute the expressions into the equation and simplify. After simplifying, the equation becomes \(-Y^2 + 3\sqrt{2}X = 0\). This is now free of cross-product terms.

Step 5: Translate origin if needed

Since our transformed equation \(-Y^2 + 3\sqrt{2}X = 0\) doesn't require further translation to complete the square or change the form, we proceed to graphing. This equation resembles \(-Y^2 = -3\sqrt{2}X\), indicating a parabola with the axis along the X-axis in the transformed coordinate system.

Step 6: Draw the graph

Plot the transformed equation \(-Y^2 + 3\sqrt{2}X = 0\). Reflect on the significance of each parameter and feature on the graph, indicating the new X and Y axes. This is a parabola opening in the direction of the positive X-axis in the rotated coordinate system.

Key Concepts

Rotation of Axes

In the world of conic sections, the rotation of axes is a transformative process that helps to simplify equations. This process is particularly useful when dealing with conic sections that have a cross-product term, such as the term \(7xy\) in our given equation.
The key to rotation of axes lies in the angle \(\theta\), which we calculate to eliminate the cross-product term. The formula used is \(\tan(2\theta) = \frac{B}{A-C}\), where \(A, B,\) and \(C\) are coefficients from the general quadratic form. Our equation yielded \(\tan(2\theta) = \text{undefined}\), leading us to conclude that \(\theta = \frac{\pi}{4}\). This means a 45-degree rotation is needed.
When rotating the axes, new variables \(X\) and \(Y\) are introduced using:

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

Through substitution, the expression for our original variables using these new coordinates eliminates the troublesome cross-product term, simplifying the relationship between our variables.

Elimination of Cross-product

The presence of a cross-product term like \(7xy\) in conic equations can complicate the analysis and graphing of these shapes. To eliminate this term, we use the rotation of axes.
Once the angle for rotation was determined as \(\theta = \frac{\pi}{4}\), we substituted our rotated axis variables \(x\) and \(y\) with expressions in terms of new variables \(X\) and \(Y\). This made our original equation simpler:

• \(x = \frac{X-Y}{\sqrt{2}}\)
• \(y = \frac{X+Y}{\sqrt{2}}\)

After substituting these into the equation and simplifying, the result was an equation free from cross-product terms: \(-Y^2 + 3\sqrt{2}X = 0\). This step is critical as it transforms a complex conic into a more recognizable form that is easier to analyze and interpret, setting the stage for further manipulation or graphing.

Equation of a Parabola

The equation \(-Y^2 + 3\sqrt{2}X = 0\) is the result of simplifying and rotating our original equation. Recognizing this form is important because it indicates a parabola.
To understand the structure of this parabola:

• The absence of a cross-product term already simplifies our understanding of its orientation.
• The equation can be rewritten as \(-Y^2 = -3\sqrt{2}X\), showing it opens to the right.

This equation informs us that our parabola is horizontally oriented along the transformed X-axis. The graphical representation will show a parabola that opens in the positive X direction according to the rotated axes.
Parabolas are symmetric and have distinct vertex and axis alignments. Understanding these features allows us to better grasp the geometric properties and potential transformations needed during analysis and graphing.