Problem 50 - College Algebra

Question

For the following exercises, determine the angle of rotation in order to eliminate the xy term. Then graph the new set of axes. $$6 x^{2}-5 \sqrt{3} x y+y^{2}+10 x-12 y=0$$

Step-by-Step Solution

Answer

The angle of rotation to eliminate the \(xy\) term is \(60^\circ\).

Step 1: Identify the coefficients

From the given equation \(6x^2 - 5\sqrt{3}xy + y^2 + 10x - 12y = 0\), identify the coefficients \(A\), \(B\), and \(C\) for the quadratic terms: \(A = 6\), \(B = -5\sqrt{3}\), and \(C = 1\).

Step 2: Find the angle of rotation formula

Use the formula to eliminate the \(xy\) term: \( \tan(2\theta) = \frac{B}{A - C} \). Substitute the values: \( \tan(2\theta) = \frac{-5\sqrt{3}}{6 - 1}\).

Step 3: Solve for \(\theta\)

Calculate \( \tan(2\theta) = \frac{-5\sqrt{3}}{5} = -\sqrt{3} \). This gives \( 2\theta = 120^\circ \) or \( 300^\circ \). Therefore, \( \theta = 60^\circ \) or \( 150^\circ \). Since we're looking for an acute angle, choose \( \theta = 60^\circ \).

Step 4: Recalculate axes using rotation angle

Apply the rotation matrix to transform original coordinates (x, y) to new coordinates (x', y') using \( x = x'\cos(60^\circ) - y'\sin(60^\circ) \) and \( y = x'\sin(60^\circ) + y'\cos(60^\circ) \) to replace in the original equation.

Step 5: Graph the new axes

Visualize the graph by plotting the transformed equation in the (x', y') coordinate system. The (x', y') axes are rotated 60° counterclockwise relative to the original (x, y) axes.

Key Concepts

Quadratic Equation

A quadratic equation in two variables, such as this one, can often appear in the form of \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\). It involves polynomial expressions of degree two. These are the equations that can graphically represent conic sections, including ellipses, circles, parabolas, and hyperbolas.

• The coefficient \(A\) is associated with the \(x^2\) term.
• The coefficient \(B\) is the factor of the \(xy\) term.
• The coefficient \(C\) corresponds to the \(y^2\) term.

The solution process often involves manipulating the equation to achieve a form that is easier to analyze or graph,such as by eliminating the \(xy\) term. The presence of the \(Bxy\) term indicates a rotated conic section,which can be realigned by determining an angle of rotation.

Eliminate xy Term

The \(xy\) term in a quadratic equation can complicate the equation's graphing and interpretation.By eliminating this term, the equation becomes simpler and represents a standard conic section.To remove the \(xy\) term, use the trigonometric identity for a rotated coordinate system.The formula \( \tan(2\theta) = \frac{B}{A - C} \) helps find the angle of rotation \(\theta\).

• Calculate \( \tan(2\theta)\) by substituting the coefficients \(B, A,\) and \(C\).
• In this example, \( \tan(2\theta) = \frac{-5\sqrt{3}}{5} = -\sqrt{3}\).
• This calculation resolves to \(2\theta = 120^\circ\) or \(300^\circ\).

We choose the acute angle \(\theta = 60^\circ\) for further simplification.

Coordinate Transformation

Coordinate transformation refers to converting the coordinates of points from one system to another.In this context, it is used to align the axes according to the angle of rotation found.Apply the rotation matrix to the coordinates \((x, y)\) to get new coordinates, \((x', y')\).The transformation formulas are:

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

For \(\theta = 60^\circ\), substitute the trigonometric values:

• \(\cos(60^\circ) = \frac{1}{2}\)
• \(\sin(60^\circ) = \frac{\sqrt{3}}{2}\)

This substitution into the formulas modifies the equation so that the \(xy\) term disappears.

Graphing Axes Rotation

Graphing the axes after a rotation can help visualize the effect of eliminating the \(xy\) term.The new set of axes, \((x', y')\), is rotated \(60^\circ\) counterclockwise from the original \((x, y)\) axes.When graphing, ensure:

• The origin remains fixed.
• The axes are rotated without altering distances between points.
• New grid lines reflect the rotation angle, making visual interpretation easier.

This rotation aligns the conic section with the transformed axes, simplifying the structure of the graph.Such visual aids provide insight into the behavior and geometry of the equation's solutions.