Problem 18 - Precalculus with Limits

Question

In Exercises 13-26, rotate the axes to eliminate the \(xy\)-term in the equation. Then write the equation in standard form. Sketch the graph of the resulting equation, showing both sets of axes. \(2x^2 -3xy-2y^2+10 = 0\)

Step-by-Step Solution

Answer

The equation after rotation and written in standard form is \(\frac{{x'}^2}{\frac{5}{3}} - \frac{{y'}^2}{10} = 1\). It represents a hyperbola which opens horizontally with vertices along the x'-axis, and is rotated counterclockwise by angle θ from the original axes.

Step 1: Calculation of Angle of Rotation

The first step is to calculate the angle of rotation, using the formula \( \frac{-b}{a-c}\), where a is 2 (coefficient of \(x^2\)), b is -3 (coefficient of \(xy\)), and c is -2 (coefficient of \(y^2\)). Plug in these values to get \( \frac{- (-3)}{2 -(-2)} \) which simplifies to 0.75. Therefore, \(2θ = tan^{-1}(0.75)\) where \(θ\) is the angle of rotation.

Step 2: Transformation of Variables

The next step is to conduct a transformation of variables via a rotation of axes matrix. Let \(x'\) be \(x cos θ - y sin θ \) and \(y'\) be \( x sin θ + y cos θ \). Substitute these terms in place of x and y in the equation.

Step 3: Substitution in the Equation

In this step, plug the variables \(x'\) and \(y'\) into the given equation. This will help eliminate the \(xy\)- term. After substitution, simplify the equation to get \( 6{x'}^2 - {y'}^2 +10 = 0 \)

Step 4: Write the Equation in Standard Form

Dividing the equation \( 6{x'}^2 - {y'}^2 +10 = 0 by -10\) we get \(\frac{-6{x'}^2}{10} + \frac{{y'}^2}{10} -1 = 0\), which can be rearranged to \(\frac{{x'}^2}{\frac{5}{3}} - \frac{{y'}^2}{10} = 1\). This is the equation in standard form.

Step 5: Sketch the Graph

The graph of the equation \(\frac{{x'}^2}{\frac{5}{3}} - \frac{{y'}^2}{10} = 1\) will be a hyperbola that opens horizontally with vertices along the x'-axis. It is rotated at an angle θ with respect to the original xy axes.

Key Concepts

Angle of Rotation

The angle of rotation is a fundamental concept when dealing with equations containing an \(xy\)-term that does not fit the standard conic section equations. In our example, the equation \(2x^2 -3xy-2y^2+10 = 0\) contains such a term. To eliminate this term and simplify the equation, we need to rotate the coordinate system by an angle where this \(xy\)-term vanishes.

The formula to calculate the angle of rotation here is \(\tan(2\theta) = \frac{-b}{a-c}\), where \(a\), \(b\), and \(c\) are coefficients from our equation. For our example, this calculation equates to \(\tan(2\theta) = \frac{3}{4}\), leading to an angle \(\theta\) which will be used in the process of rotating the axes. It is essential to find this angle precisely as it ensures the simplification of the equation into a more manageable form.

Transformation of Variables

After determining the angle of rotation, the next step involves a transformation of variables. This step is crucial as it transitions us from the original axes to the new, rotated axes. Instead of using \(x\) and \(y\), we will now use \(x'\) and \(y'\).

To define \(x'\) and \(y'\), we use the rotation matrix coupled with our original variables, that is \({x' = x \cos(\theta) - y \sin(\theta)}\) and \({y' = x \sin(\theta) + y \cos(\theta)}\). These new variables \(x'\) and \(y'\) help us to restate the original equation in a way that the \(xy\)-term is eliminated, leading to a recognizable conic section equation.

Standard Form of Conic Sections

Once we have transformed the variables, it is often necessary to rewrite the equation in the standard form of conic sections. The standard form makes it much easier to analyze and graph the equation. For hyperbolas, the standard form is \(\frac{x'^2}{a^2} - \frac{y'^2}{b^2} = 1\), where \(^2\) and \(^2\) are squares of the lengths of the axes of the hyperbola.

In our transformed equation from earlier, \(6x'^2 - y'^2 +10 = 0\) can be rearranged into \(\frac{x'^2}{5/3} - \frac{y'^2}{10} = 1\), to resemble the standard form. This explicit form helps us to identify the characteristics of the conic section, such as the axes, vertices, and foci, facilitating a simpler analysis and sketching.

Graph Sketching

Graph sketching is a powerful tool for visualizing equations. After obtaining the standard form of the equation, we can sketch the graph to better understand the geometric properties of our conic section. In this context, our hyperbola \(\frac{x'^2}{5/3} - \frac{y'^2}{10} = 1\) indicates it opens horizontally due to the positive \(x'^2\)-term.

We'll show both the original and the rotated axes while drawing the graph for clarity. Plot the vertices along the \(x'\)-axis, taking into account the values derived from the standard form. In our example, the graph of the rotated hyperbola would illustrate the effects of transforming through rotation and allow us to clearly see the difference in orientation between the new and the original axes.