Problem 29 - 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. \(x^{2}+4 x y+y^{2}-2 x+1=0\)

Step-by-Step Solution

Answer

Rotate by \(45^\circ\); new equation is \((x')^2 + (y')^2 - \sqrt{2}(x' + y') + 1 = 0\).

Step 1: Identify Rotation Formulas

Recognize that to eliminate the \(xy\) term from the equation \(x^2 + 4xy + y^2 - 2x + 1 = 0\), we need the rotation transformation formulas: \(x = x' \cos \theta - y' \sin \theta\) and \(y = x' \sin \theta + y' \cos \theta\).

Step 2: Find the Rotation Angle Formula

To eliminate the \(xy\) term using the angle \(\theta\), use the formula \(\tan 2\theta = \frac{B}{A-C}\), where \(A = 1\), \(B = 4\), and \(C = 1\) for the given equation.

Step 3: Calculate \(\tan 2\theta\)

Substitute \(A = 1\), \(B = 4\), and \(C = 1\) into the formula: \[ \tan 2\theta = \frac{4}{1 - 1} = \frac{4}{0}. \] The formula yields an undefined value, indicating \(2\theta = 90^\circ\), thus \(\theta = 45^\circ\).

Step 4: Rewrite the Equation Using Rotation

Use \(\theta = 45^\circ\) to transform coordinates: \(x = \frac{x' - y'}{\sqrt{2}}\) and \(y = \frac{x' + y'}{\sqrt{2}}\). Substitute these into the original equation \(x^2 + 4xy + y^2 - 2x + 1 = 0\).

Step 5: Simplify the New Equation

After substituting and simplifying, the transformed equation in \(x'\) and \(y'\) without the \(x'y'\) component is \((x')^2 + (y')^2 - \sqrt{2}(x' + y') + 1 = 0\).

Key Concepts

Angle Determination

Determining the angle of rotation is crucial when transforming the coordinates of a given equation to eliminate cross terms like \(xy\). When faced with an equation containing an \(xy\) term, such as \(x^2 + 4xy + y^2 - 2x + 1 = 0\), the goal is to find an angle \(\theta\) that allows us to rewrite the equation without the cross term. To achieve this, we rely on a specific trigonometric relationship: \(\tan 2\theta = \frac{B}{A-C}\). Here, \(A\), \(B\), and \(C\) refer to the coefficients from the quadratic terms in the equation. For our example:

• \(A = 1\)
• \(B = 4\)
• \(C = 1\)

Plugging these into our formula yields an undefined value due to \(\frac{4}{0}\), signaling that \(2\theta = 90^\circ\). Consequently, \(\theta = 45^\circ\), effectively guiding the coordinate transformation for elimination of the \(xy\) term.

Coordinate Transformation

Coordinating transformation in mathematics repositions the axes to simplify equations, thereby removing unwanted terms like \(xy\) from the equation. By applying this to our equation, we achieve a clearer understanding. Given \(\theta = 45^\circ\), we proceed to transform our coordinates using rotational formulas. These formulas are:

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

Since \(\theta = 45^\circ\), \(\cos 45^\circ = \sin 45^\circ = \frac{1}{\sqrt{2}}\), leading us to simplify to:

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

Using these formulas, we replace \(x\) and \(y\) in the original equation, seamlessly shifting our system to \(x'\) and \(y'\) coordinates, preparing us for further simplifications.

Elimination of Cross Terms

The primary aim when performing a rotation of axes on a conic section equation is to eliminate the cross term \(xy\). Successfully identifying \(\theta = 45^\circ\) guides us in the coordinate transformation process. Substituting the new coordinates into the equation, we replace \(x\) and \(y\) and open up the potential to simplify the equation without \(x'y'\).Here is the process briefly:

• Substitute \(x = \frac{x' - y'}{\sqrt{2}}\) and \(y = \frac{x' + y'}{\sqrt{2}}\) into the original equation \(x^2 + 4xy + y^2 - 2x + 1 = 0\).
• Simplify the resulting terms using algebraic skills to clear out the \(x'y'\) term.

After the application of this transformation, the equation reduces to \((x')^2 + (y')^2 - \sqrt{2}(x' + y') + 1 = 0\), where the intrusive \(x'y'\) term has been skillfully eliminated.

Conic Sections

Conic sections are shapes formed by the intersection of a plane with a double-napped cone. Commonly, these include circles, ellipses, parabolas, and hyperbolas. Equations defining conic sections can become complicated, especially with the presence of an \(xy\) term indicating a rotated conic. To simplify these complex equations, especially for easier graphing and analysis, it's essential to transform the equation to align with the primary axes. This transformation via rotation eliminates the \(xy\) term, as seen with our example. In absence of the \(xy\) component, these sections reveal their standard form, allowing easier identification of fundamental characteristics like the center, axes, and orientation. This clarity aids in real-world applications like physics and engineering, where rotational transformations provide elegant solutions to more complex geometrical problems.