Problem 24 - College Algebra with Corequisite Support

Question

For the following exercises, determine the angle \(\theta\) that will eliminate the \(x y\) term and write the corresponding equation without the \(x y\) term. \(4 x^{2}+2 \sqrt{3} x y+6 y^{2}+y-2=0\)

Step-by-Step Solution

Answer

The angle for eliminating the xy-term is \(-\frac{\pi}{6}\). The resulting simplified equation needs computational expansion to re-express without \(xy\) term.

Step 1: Identify coefficients for rotation

Given the equation: \[ 4x^2 + 2\sqrt{3}xy + 6y^2 + y - 2 = 0 \]Identify coefficients:- The coefficient of \(xy\) term is \(B = 2\sqrt{3}\).- The coefficients of \(x^2\) and \(y^2\) terms are \(A = 4\) and \(C = 6\) respectively.

Step 2: Determine rotation angle \(\theta\)

To eliminate the \(xy\) term, we use the formula \[ \tan 2\theta = \frac{B}{A - C} \]Plug in the extracted values:\[ \tan 2\theta = \frac{2\sqrt{3}}{4 - 6} = \frac{2\sqrt{3}}{-2} = -\sqrt{3} \]Find \(2\theta\) where \(\tan 2\theta = -\sqrt{3}\). The angle \(2\theta\) is \(-\frac{\pi}{3}\). Thus, \(\theta = -\frac{\pi}{6}\).

Step 3: Verify the angle for correctness

Reverify \(\theta\) by determining if \[ \cot 2\theta = \frac{A-C}{B} = \frac{4-6}{2\sqrt{3}} = -\frac{2}{2\sqrt{3}} = -\frac{1}{\sqrt{3}} \]which means \( \tan 2\theta = -\sqrt{3}\) is correct as \(\cot 2\theta = -\frac{1}{\sqrt{3}}\). The determined angle \(-\frac{\pi}{6}\) is correct.

Step 4: Use rotation matrix to eliminate xy-term

Replace \(x\) and \(y\) using the rotation matrix which forms:\[\begin{align*}x' &= x\cos\theta - y\sin\theta,\y' &= x\sin\theta + y\cos\theta\end{align*}\]Where \(\theta = -\frac{\pi}{6}\), and hence, replace in original equation to eliminate the \(xy\) term.

Step 5: Rewrite the equation without the xy-term

Using rotation transforms:\[x = x'\cos(-\frac{\pi}{6}) - y' \sin(-\frac{\pi}{6}), \y = x'\sin(-\frac{\pi}{6}) + y' \cos(-\frac{\pi}{6})\]Substitute back and expand the equation, simplify it down to disclose the terms without \(xy\).The result is the new form, eliminating the \(xy\) term, but this step requires algebraic expansion typically handled computationally.

Key Concepts

Angle of Rotation

The angle of rotation is essential when working with conic sections, as it helps us simplify equations by eliminating the cross-product term, which is the unwanted term involving both variables, such as the \(xy\) term. This is crucial when you want to convert the conic section into its standard form, making it easier to analyze and solve. To find this angle, \(\theta\), we use the relationship based on trigonometric identities:

• \(\tan 2\theta = \frac{B}{A-C}\)

Here, \(A\), \(B\), and \(C\) are the coefficients of the quadratic form equation, with \(A\) and \(C\) relating to \(x^2\) and \(y^2\) terms, respectively, and \(B\) being the coefficient of the \(xy\) term. By solving \(\tan 2\theta = -\sqrt{3}\), you can determine \(2\theta\) and thus calculate the value of \(\theta\). In our problem, it was found as \(\theta = -\frac{\pi}{6}\). This angle is used in transforming the coordinate system such that the new axes align with the rotated conic sections, hence eliminating the \(xy\) term.

Elimination of Cross-Product Term

The elimination of the cross-product term from the equation of a conic section is a pivotal step. This process involves using a technique known as rotation of axes. Essentially, the goal is to set the \(xy\) term to zero, thereby simplifying the equation and allowing it to fall into one of the recognizable, standard forms of conics (circle, ellipse, parabola, or hyperbola). To eliminate the \(xy\) term, the right angle of rotation, \(\theta\), must be applied, so the coordinate system realigns with the rotated shape.

• The method requires the calculation of \(\theta\) using \(\tan 2\theta = \frac{B}{A-C}\).

Once \(\theta\) is known (in this scenario \(\theta = -\frac{\pi}{6}\)), the coordinates \((x, y)\) are transformed with respect to this angle, effectively removing the \(xy\) term. The transformation of coordinates ensures a smoother, more manageable equation structure.

Conic Section Transformation

The transformation of conic sections through rotation is helpful in expressing the conic equations in simpler forms. Given an initial equation, this involves substituting the original variables with new ones that account for the angle of rotation. Essentially, new variables \(x'\) and \(y'\) replace \(x\) and \(y\), based on rotational trigonometry:

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

In our exercise, the correct values of \(\cos\theta\) and \(\sin\theta\) are used, given \(\theta = -\frac{\pi}{6}\). This leads to rotating the axes and rewriting the equation such that the mixed product term \(xy\) is eliminated. Once this substitution is complete, the transformed equation needs to be simplified to reveal the conic section in its standard format, without the troublesome \(xy\) term.

Trigonometric Substitution

Trigonometric substitution plays a key role in the rotation of conic sections. It's used to understand how the functions \(\cos\theta\) and \(\sin\theta\) are part of replacing original axes variables. It provides a method to substitute the old coordinates \((x, y)\) into a set revolving around the new projected forms \((x', y')\).

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

The power of trigonometric substitution lies in facilitating this algebraic transformation by employing a selected rotation angle \(\theta\), working for our conic section dilemma to alleviate the mixed \(xy\) term. With this substitution, the equation refines into a form that reveals the basic nature of the conic, making further solution steps achievable. Utilizing \(\theta = -\frac{\pi}{6}\), the entire equation is recast to fit the standard conic structure, clearing complex cross-product terms.