Problem 29 - College Algebra

Question

For the following exercises, determine the angle ? that will eliminate the xy term and write the corresponding equation without the \(xy\) term. $$x^{2}+4 x y+y^{2}-2 x+1=0$$

Step-by-Step Solution

Answer

The angle \(\theta\) is \(\frac{\pi}{4}\). The transformed equation, free from the \(xy\) term, is obtained after substituting x and y with the rotated coordinates.

Step 1: Identify the coefficients

The given quadratic equation is \(x^{2}+4xy+y^{2}-2x+1=0\). Here, the coefficient of \(xy\) is 4, \(a=1\) for \(x^2\), \(b=4\) for \(xy\), and \(c=1\) for \(y^2\). The coefficient terms for the rotation transformation are \(a\), \(b/2\), and \(c\). We need these to find the angle \(\theta\).

Step 2: Use the angle formula

The formula to find the angle \(\theta\) to eliminate the \(xy\) term is \( \theta = \frac{1}{2} \arctan\left(\frac{b}{a-c}\right) \). Substituting the given values: \( \theta = \frac{1}{2} \arctan\left(\frac{4}{1-1}\right) \). Since \(a - c = 0\), the formula becomes \( \theta = \frac{1}{2} \frac{\pi}{2} = \frac{\pi}{4} \).

Step 3: Transform the coordinates

To eliminate the \(xy\) term, use the coordinate transformation equations which include: \(x' = x \cos \theta + y \sin \theta\) and \(y' = -x \sin \theta + y \cos \theta\). For \(\theta = \frac{\pi}{4}\), \(\cos \theta = \sin \theta = \frac{\sqrt{2}}{2}\). Thus, \(x' = \frac{\sqrt{2}}{2}x + \frac{\sqrt{2}}{2}y\) and \(y' = -\frac{\sqrt{2}}{2}x + \frac{\sqrt{2}}{2}y\).

Step 4: Substitute the transformed coordinates

Replace \(x\) and \(y\) in the given equation with \(x'\) and \(y'\) using the transformations: \(x = \frac{1}{\sqrt{2}}(x'+ y')\) and \(y = \frac{1}{\sqrt{2}}(-x'+ y')\). Substitute these into the original equation \(x^2 + 4xy + y^2 - 2x + 1 = 0\).

Step 5: Simplify the new equation

After substitution, calculate and simplify the terms in the equation. This involves expanding \((x')\) and \((y')\) terms, and simplifying based on trigonometric and arithmetic rules. The substitution simplifies and eliminates the \(xy\) term, leaving an equation in terms of \(x'^2\) and \(y'^2\) only. After all simplifications, you will get an equation like \(x'^2 + kx' + ly' + m = 0\) with no \(x'y'\) term.

Key Concepts

Conic Sections

Conic sections are the curves obtained by intersecting a plane with a cone. These curves include circles, ellipses, parabolas, and hyperbolas. Each type has distinct characteristics:

• Circles and ellipses: Formed when the intersecting plane is tilted but does not touch both nappes (the upper and lower parts of the cone).
• Parabolas: Created when the plane is parallel to the slope of the cone.
• Hyperbolas: Occur when the plane intersects both nappes of the cone.

Understanding these forms is crucial for learning about the geometry behind different shapes. Conic sections can be represented by quadratic equations. The form of these equations helps us categorize them into types, as done in the given problem by identifying coefficients and recognizing them as part of a quadratic equation forming a conic section.
Identifying the type of conic section involved allows us to apply different transformations, like rotation of axes, to simplify equations and analyze these shapes more effectively.

Coordinate Transformation

Coordinate transformation involves changing the coordinate system to simplify mathematical equations. For example, by rotating the coordinate axes, we can eliminate the cross-product term (like the \(xy\) term in a quadratic equation). This procedure is known as rotation of axes.
In mathematical terms, the new coordinates are found by:

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

Here, \(\theta\) is the angle of rotation. For the specific equation \(x^2 + 4xy + y^2 - 2x + 1 = 0\), a \(45^\circ\) (or \(\frac{\pi}{4}\)) rotation simplifies the equation by removing the \(4xy\) term.
Transformations like these are powerful tools in making complex equations more understandable and more straightforward to solve. By transforming coordinates, we can change the perspective from which we view a problem, often revealing simpler relationships or solutions.

Quadratic Equations

Quadratic equations involve terms up to the second degree and often appear in various forms, such as the standard form \(ax^2 + bxy + cy^2 + dx + ey + f = 0\). Involving two variables, they define conic sections discussed earlier.
Solving quadratic equations can sometimes require changing the form or re-arranging terms. Using transformations might simplify the problem by removing certain unwanted terms, like the \(xy\) term in our exercise. This is achieved through techniques like completing the square or using specific transformations like the rotation used in this exercise.
In this context, the main goal is to identify ways to simplify the equation into a recognizable form, revealing the nature of the conic section or solving for variables. This highlights the importance of balancing different approaches to solving quadratic equations effectively, including algebraic manipulations and geometric insights.