Problem 52 - College Algebra with Corequisite Support

Question

For the following exercises, determine the angle of rotation in order to eliminate the xy term. Then graph the new set of axes. \(6 x^{2}-8 \sqrt{3} x y+14 y^{2}+10 x-3 y=0\)

Step-by-Step Solution

Answer

The angle of rotation to eliminate the xy term is 30 degrees.

Step 1: Identify the coefficients in the equation

Identify the coefficients from the given quadratic equation: \(6x^{2} - 8\sqrt{3} xy + 14y^{2} + 10x - 3y = 0\). Here, \(A = 6\), \(B = -8\sqrt{3}\), and \(C = 14\).

Step 2: Use the rotation formula to find tan(2θ)

For an equation \(Ax^2 + Bxy + Cy^2\), the angle \(\theta\) that removes the \(xy\) term is found using \( \tan(2\theta) = \frac{B}{A - C} \). Plug in the values: \( \tan(2\theta) = \frac{-8\sqrt{3}}{6 - 14} = \frac{-8\sqrt{3}}{-8} = \sqrt{3} \).

Step 3: Solve for θ using the tangent identity

Since \(\tan(2\theta) = \sqrt{3}\), we need to find \(2\theta\) where tangent value is \(\sqrt{3}\). This corresponds to \(2\theta = 60^{\circ}\), as \(\tan(60^{\circ}) = \sqrt{3}\). Thus, \(\theta = 30^{\circ}\).

Step 4: Graph the rotation of axes

To graph the rotation of axes by an angle \(\theta = 30^{\circ}\), draw the new \(x'y'\)-axes rotated from the original \(xy\)-axes by 30 degrees. The new axes will intersect the origin and make an angle of 30 degrees with the old axes.

Key Concepts

Rotation of Axes

When dealing with quadratic equations, we sometimes encounter an \(xy\) term that complicates the equation. To simplify it, we can perform a geometric transformation known as the rotation of axes. This process involves rotating the entire coordinate system around the origin by a specific angle \(\theta\).
By choosing the right angle, we can eliminate the \(xy\) term, making it easier to analyze and graph the equation. The new axes \((x', y')\) replace the original \((x, y)\), allowing us to work with a simpler form of the equation.

Angle of Rotation

To find the angle \(\theta\) for rotating the axes and eliminating the cross-product term, we utilize the identity \( \tan(2\theta) = \frac{B}{A - C} \).
In our given problem, the coefficients are \(A = 6\), \(B = -8\sqrt{3}\), and \(C = 14\). Substituting into the formula gives \( \tan(2\theta) = \sqrt{3} \).
We know that \(\tan(60^{\circ}) = \sqrt{3}\), hence \(2\theta = 60^{\circ}\) leading to \(\theta = 30^{\circ}\).
This calculation finds the precise angle needed to rotate the axes in such a way that \(xy\) becomes zero in the updated equation.

Eliminating Cross-Product Term

The ultimate goal of the rotation is to eliminate the cross-product term \(xy\) from the quadratic equation. Why do we do this? It's because having an \(xy\) term increases complexity in graphing and understanding the conic sections described by the equation.
By computing the correct angle \(\theta\) and rotationally transforming the axes by this angle, we can rewrite the equation without the \(xy\) term.
This transformation leads to simpler mathematical analysis and graphing, as the conic section (like ellipse, parabola, or hyperbola) becomes aligned with the newly oriented coordinate axes.

Graphing Equations

Once the rotation of axes is complete and the equation is simplified, it is time to graph the equation. With an angle \(\theta = 30^{\circ}\), the new \(x'\) and \(y'\) axes are inclined 30 degrees from the original axes.
Start by marking the origin point. From there, draw rotated lines in the direction of 30 degrees from the positive \(x\) and \(y\) axes.
These new lines act as the \(x'\)-axis and \(y'\)-axis. With the equation simplified, mapping the graph onto these new axes will reflect the underlying geometric shape, be it a circle, ellipse, or other conic shape.