Problem 51 - College Algebra

Question

For the following exercises, determine the angle of rotation in order to eliminate the xy term. Then graph the new set of axes. $$6 x^{2}-5 x y+6 y^{2}+20 x-y=0$$

Step-by-Step Solution

Answer

Rotate the axes by 45 degrees to eliminate the xy term.

Step 1: Write the General Form

The given equation is \( 6x^2 - 5xy + 6y^2 + 20x - y = 0 \). This is a quadratic equation in terms of \(x\) and \(y\), and we need to eliminate the \(xy\) term.

Step 2: Identify Coefficients

The coefficients we need from the equation are: \(A = 6\), \(B = -5\) (the coefficient of \(xy\)), and \(C = 6\). These coefficients correspond to terms in the equation \(Ax^2 + Bxy + Cy^2\).

Step 3: Calculate the Angle of Rotation

The formula to find the angle \(\theta\) to eliminate the \(xy\) term is given by \(\tan(2\theta) = \frac{B}{A-C}\). Substituting \(A = 6\), \(B = -5\), and \(C = 6\) into the formula, we get: \[ \tan(2\theta) = \frac{-5}{6 - 6} = \text{undefined} \]. This means \(2\theta\) corresponds to an angle of 90 degrees and \(\theta = 45\) degrees.

Step 4: Rotate the Axes

Since \(\theta = 45\) degrees, the rotation of the axes is by 45 degrees to eliminate the \(xy\) term. The transformation equations for a rotation by 45 degrees are: \(x = x'\cos(45^{\circ}) - y'\sin(45^{\circ})\) and \(y = x'\sin(45^{\circ}) + y'\cos(45^{\circ})\).

Step 5: Express in New Variables

Substitute the transformation equations into the original equation to express it in terms of \(x'\) and \(y'\). The equation becomes quadratic in \(x'^2\) and \(y'^2\) without the \(x'y'\) term, verifying the rotation was successful.

Step 6: Graph the New Set of Axes

The new set of axes, \(x'\) and \(y'\), are the result of a 45-degree rotation of the original axes. Graphically represent the transformation and verify that the \(x'y'\) term has been eliminated visually by aligning the axes with the ellipse's principal axes.

Key Concepts

Elimination of xy Term

When dealing with quadratic equations that have an \(xy\) term, it can be helpful to eliminate this term to simplify the equation. This simplification is done using a mathematical tool called the angle of rotation. The angle of rotation helps us reorient the coordinate system in such a way that the \(xy\) term disappears. Essentially, this means

• identifying the angle that allows for a clean transition from old axis to new axis
• using trigonometric formulas to adjust the equation accordingly

For the equation given, we start by recognizing the coefficients related to the \(xy\) term. Using the formula \(\tan(2\theta) = \frac{B}{A-C}\), we find the appropriate rotation angle. In this specific problem, \(2\theta\) results in 90 degrees, leaving \(\theta\) as 45 degrees. This indicates a 45-degree reorientation is necessary to eliminate the cross-product term.

Quadratic Equations

Quadratic equations are mathematical expressions that involve terms like \(Ax^2, Bxy,\) and \(Cy^2\). They are generally represented in the form \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\). In these equations, the presence of the \(xy\) term indicates a degree of complexity that can describe conic sections such as ellipses and hyperbolas. Quadratic equations:

• define curves and shapes in the coordinate plane
• can be rotated or adjusted by modifying the coefficients

For example, the given equation \(6 x^{2}-5 x y+6 y^{2}+20 x-y=0\) includes multiple terms, with \(B = -5\) corresponding to the \(xy\) term, introducing a "mixed" component or tilt to the curve in its graphical representation.

Coordinate Transformation

Coordinate transformation allows us to shift and rotate the axes for a better representation and simplification of equations. This technique uses trigonometric formulas to map coordinates from one system

• (the original axes \(x\) and \(y\)) to another new set of axes \(x'\) and \(y'\)

The equations for a 45-degree rotation are \(x = x'\cos(45^{\circ}) - y'\sin(45^{\circ})\) and \(y = x'\sin(45^{\circ}) + y'\cos(45^{\circ})\). By substituting these into the original quadratic equation, we simplify the equation's structure, removing the \(xy\) term. This transformation not only simplifies calculations but also enhances visualization, allowing for clearer analysis of shapes and properties.

Graphing Transformed Axes

Graphing after coordinate transformation displays the newly oriented axes, highlighting changes in orientation while eliminating specific complex terms like the \(xy\) term. This phase involves

• plotting the redefined axes \(x'\) and \(y'\) on a coordinate plane
• demonstrating the new orientation of the quadratic shape

In our case, after a 45-degree rotation, the equation becomes easier to graph and analyze. The \(x'^2\) and \(y'^2\) terms can be graphed effectively with the new axes aligning with the principal axes of the conic section, be it an ellipse or a hyperbola. This essentially ensures that the transformed graph represents the solution in the simplest and most understandable form.