Problem 18 - College Algebra with Corequisite Support

Question

For the following exercises, find a new representation of the given equation after rotating through the given angle. \(3 x^{2}+x y+3 y^{2}-5=0, \theta=45^{\circ}\)

Step-by-Step Solution

Answer

The rotated equation is \( x'^2 = \frac{5}{2} \).

Step 1: Recognize Coordinate Transformation

When rotating the coordinate system by an angle \( \theta \), the new coordinates \((x', y')\) can be found using the transformation formulas: \[\begin{align*}x' &= x \cos \theta + y \sin \theta, \y' &= -x \sin \theta + y \cos \theta.\end{align*}\]In this problem, \( \theta = 45^{\circ} \), so \( \cos 45^{\circ} = \sin 45^{\circ} = \frac{\sqrt{2}}{2} \).

Step 2: Substitute New Variables

Substitute \( x = \frac{\sqrt{2}}{2}(x' - y') \) and \( y = \frac{\sqrt{2}}{2}(x' + y') \) into the original equation:\[ 3x^2 + xy + 3y^2 - 5 = 0 \]By substituting for \(x\) and \(y\), our equation in terms of \(x'\) and \(y'\) can be further simplified.

Step 3: Simplify the Equation

Expand the terms to simplify the new equation:\[3 \left( \frac{\sqrt{2}}{2}(x' - y') \right)^2 + \left( \frac{\sqrt{2}}{2}(x' - y') \right)\left( \frac{\sqrt{2}}{2}(x' + y') \right) + 3 \left( \frac{\sqrt{2}}{2}(x' + y') \right)^2 - 5 = 0 \]Calculate each of these terms, factoring in constants and then adding to arrive at a simplified equation.

Step 4: Calculate and Combine Like Terms

Begin by calculating and combining the expanded terms for each component of the equation:- For \(3(x' - y')^2\) and \(3(x' + y')^2\),- For \((x' - y')(x' + y')\),Work through each multiplication and addition, carefully.The result is the equation in \((x', y')\) terms.

Step 5: Write the Final Equation

After calculations, the equation simplifies to:\[ 4x'^2 = 10 \]This is already in its simplest form, representing the rotation by \(45^\circ\). Further simplification gives:\[ x'^2 = \frac{5}{2} \]

Key Concepts

Rotation of Axes

Understanding a rotation of axes in a coordinate transformation is crucial when working with geometric equations. Rotating the axes simplifies complex expressions by aligning the form of the equation with particular geometric features like symmetry or orientation. Given the problem statement, we're rotating the equation by \( \theta = 45^{\circ} \). This particular angle is often chosen because of its symmetric properties and the simplification it offers in the transformations. The formulas used to rotate the axes incorporate basic trigonometric identities:

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

In this scenario, both \( \cos 45^{\circ} \) and \( \sin 45^{\circ} \) are equal to \( \frac{\sqrt{2}}{2} \). This makes computations simpler, as the transformation retains a mix of both \(x\) and \(y\) without favoring one overly. Rotating through this angle allows the original equation to take a new shape that's easier to work with, being a crucial initial step in our transformation journey.

Trigonometric Substitution

Trigonometric substitution is a technique used to simplify expressions containing trigonometric functions. In this case, during the coordinate transformation, you replace the coordinates \((x, y)\) with the new ones \((x', y')\), employing trigonometric identities. The success of this substitution hinges on the direct and symmetric nature of the angles involved. For our specific example, the angle of \( 45^{\circ} \) leads to substitutions that cleverly exploit the simplicity of the trigonometric values \( \cos 45^{\circ} = \sin 45^{\circ} = \frac{\sqrt{2}}{2} \).

• Substitute \( x = \frac{\sqrt{2}}{2}(x' - y') \)
• Substitute \( y = \frac{\sqrt{2}}{2}(x' + y') \)

By substituting these into the original equation, the representation shifts to a form that we can further manipulate algebraically. This swap isn't merely for convenience; it fundamentally reorients the problem, giving a fresh perspective that can reveal underlying symmetries not visible in the original form.

Algebraic Manipulation

Once we've successfully rotated the axes and swapped variables using trigonometric substitution, we proceed to algebraic manipulation. The goal here is to rework the equation to a clearer and more solvable form. This process involves expanding terms and applying algebraic operations such as distribution: When dealing with expressions like \(3(x'-y')^2\) and \((x'-y')(x'+y')\), each term must be expanded following standard algebraic rules:

• Expand \( (x'-y')(x'+y') \) to get \( x'^2 - y'^2 \)
• Calculate \( 3(x'-y')^2 = 3(x'^2 - 2x'y' + y'^2) \)
• Sum similar terms to simplify the equation

Careful to compute each multiplication and addition step-by-step ensures accuracy and simplifies the original complexity. Ultimately, these manipulative steps pave the way to a transformed equation that represents our problem's rotated state.

Quadratic Equation Transformation

The linear and quadratic terms in an equation often need simplification through transformation techniques, especially when rotating axes. After expanding and simplifying through algebraic manipulation, we are often left with quadratic forms or linear terms. This transformation is essential to extract useful geometric or algebraic information.In the given example, the equation simplifies down to a quadratic form that is easy to analyze:The final form attained is \( 4x'^2 = 10 \), simplifying further into \( x'^2 = \frac{5}{2} \). This represents the simplest form of the original equation after rotation.Quadratic equation transformation helps in seeing the geometric relationship within the transformed space. It takes an original equation and leaves us with information-bound forms like parabolas, ellipses, or hyperbolas. In our case, it's a shrunk pivot of a circle or ellipse as indicated by the quadratic term \( x'^2 \), showing how effective transformations can be.