Problem 42 - College Algebra

Question

For the following exercises, graph the equation relative to the \(x^{\prime} y^{\prime}\) system in which the equation has no \(x^{\prime} y^{\prime}\) 'term. $$4 x^{2}-3 \sqrt{3} x y+y^{2}-22=0$$

Step-by-Step Solution

Answer

Rotate the graph by 30° to eliminate the \(xy\) term.

Step 1: Identify the Original Equation

The given equation in terms of the original axes is \( 4x^2 - 3\sqrt{3}xy + y^2 - 22 = 0 \). Our task is to identify the rotation needed to eliminate the \( xy \) term.

Step 2: Find Rotation Angle

To eliminate the \( xy \) term, determine the angle of rotation \( \theta \) using the formula \( \tan(2\theta) = \frac{-B}{A - C} \). Here, \( A = 4 \), \( B = -3\sqrt{3} \), and \( C = 1 \), so \( \tan(2\theta) = \frac{3\sqrt{3}}{3} = \sqrt{3} \). This gives \( 2\theta = 60^\circ \) and \( \theta = 30^\circ \).

Step 3: Calculate Transformation Equations

The equations for the transformation are \( x = x'\cos\theta - y'\sin\theta \) and \( y = x'\sin\theta + y'\cos\theta \). Substituting \( \theta = 30^\circ \), we have \( \cos\theta = \frac{\sqrt{3}}{2} \) and \( \sin\theta = \frac{1}{2} \), so the equations become \( x = \frac{\sqrt{3}}{2} x' - \frac{1}{2} y' \) and \( y = \frac{1}{2} x' + \frac{\sqrt{3}}{2} y' \).

Step 4: Substitute and Simplify

Substitute the expressions for \( x \) and \( y \) into the original equation. The terms involving \( x' \) and \( y' \) will eliminate the \( x'y' \) terms, resulting in \( a{x'}^2 + c{y'}^2 - 22 = 0 \) where \( a \) and \( c \) are constants found through substitution.

Step 5: Graph the Resulting Equation

The new equation in terms of \( x' \) and \( y' \) is \( 5x'^2 - y'^2 = 22 \). This is a hyperbola centered at the origin. Use the vertices and asymptotes to sketch the graph in the \( x'y' \) coordinate system.

Key Concepts

Graphing Transformations

Transforming the graph of an equation is essential for simplifying complex expressions. When handling equations with terms that are not straightforward to interpret graphically, such as the \( xy \) term in our original equation, performing a rotational transformation can be invaluable.

By eliminating this \( xy \) term through a rotation, we transform the equation into a more recognizable form. In this exercise, the original equation was rotated to simplify it into the form of a hyperbola.

• Rotation equations are critical as they preserve the structure of curves on the graph.
• They convert coordinates from the original \((x, y)\) system to a newly oriented \((x', y')\) system.
• Understanding the rotation process helps clarify which transformations are being applied to what part of the equation.

This transformation directly impacts how the graph is drawn, making it clearer and more manageable.

Hyperbola Graphing

A hyperbola is a type of conic section that is formed by intersecting a right circular cone with a plane at an angle, creating two separate curves. In our solution, after removing the \( xy \) term, the equation simplifies to a hyperbola.

The hyperbola \( 5x'^2 - y'^2 = 22 \) is centered at the origin in \((x', y')\) coordinates. It's important to understand these properties when graphing:

• Identifying vertices: The hyperbola opens along the axis where the coefficient of the squared term is positive.
• Asymptotes serve as lines that the hyperbola approaches but never touches. For this hyperbola, they help in estimating its "openness."

Hyperbolas graph as two open curves or "branches" and understanding their shape helps when translating equations into visual graphs.

Angle of Rotation

The angle of rotation is crucial in converting an equation so that it aligns with a new coordinate system. In this exercise, we employ rotation to eliminate the \( xy \) term using the angle \( \theta \).

The formula \( \tan(2\theta) = \frac{-B}{A - C} \) is indispensable for discovering the necessary angle:

• The known values of \( A \), \( B \), and \( C \) from the equation guide us to compute \( 2\theta \) and \( \theta \).
• With \( 2\theta = 60^\circ \), we conclude \( \theta = 30^\circ \).
• This angle allows for a rotation that simplifies the equation's graphing in a transformed system.

Mastering the use of angle rotation supports clarity when transforming complex graphs and solving them effectively.