Problem 24 - Precalculus with Limits

Question

In Exercises 13-26, rotate the axes to eliminate the \(xy\)-term in the equation. Then write the equation in standard form. Sketch the graph of the resulting equation, showing both sets of axes. \(16x^2 -24xy+9y^2-60x-80y+100 = 0\)

Step-by-Step Solution

Answer

Following all the steps and simplifying the equation will result in a standard form equation. This equation can then be graphed, showing both the original and rotated axes.

Step 1: Determine the Angle of Rotation

The angle between the original axes and the new ones is represented by \(\theta\). This can be established by employing the formula: \(\tan (2 \theta) = \frac{2b}{a - c}\). From the quadratic form \(Ax^2 + 2Bxy + Cy^2 + Dx + Ey + F = 0\), we can establish that \(B = -12, A = 16\), and \(C = 9\). Utilizing these values we get \(\theta = 0.5 \arctan ( -\frac{2(-12)}{16 - 9} )\)

Step 2: Rotate the Axes

For rotating the axes, the coordinates (\(x\), \(y\)) are transformed to (\(x'\), \(y'\)) using the rotation matrix: \[x = x' \cos \theta - y' \sin \theta\] and \[y = x' \sin \theta + y' \cos \theta\]. Substitute these equations into the original given equation and simplify.

Step 3: Write Rotation Matrix

Write the equation using the rotation matrix. Using the above transformation equations and the calculated angle of rotation, the equation can be transformed into \[16(x' \cos \theta - y' \sin \theta)^2 -24(x' \cos \theta - y' \sin \theta)(x' \sin \theta + y' \cos \theta)+9(x' \sin \theta + y' \cos \theta)^2-60x'-80y'+100 = 0\]

Step 4: Simplification

After substituting the \(x\) and \(y\) with their corresponding terms in terms of \(x'\) and \(y'\), simplify the equation to its standard form.

Step 5: Sketch the graph

It is now possible to sketch the graph once the equation is in its standard form. Remember to show the original and rotated axes in the sketch. The final graph may look like an ellipse or a line depending upon the given quadratic equation.

Key Concepts

Rotation of Axes

To eliminate the awkward and complicating \(xy\)-term from a general quadratic equation, we perform a rotation of the coordinate axes. This clever trick involves rotating the entire coordinate system by a certain angle, referred to as \(\theta\), simplifying our equation in the process.
To calculate this angle of rotation, we use a specific formula derived from the coefficients of the quadratic component: \(\tan(2\theta) = \frac{2B}{A - C}\). Here, \(B\) is the coefficient of the \(xy\) term, while \(A\) and \(C\) are the coefficients of \(x^2\) and \(y^2\) respectively. For example, in our task, \(B = -12\), \(A = 16\), and \(C = 9\). Solving, we find the required angle \(\theta\) that realigns the axes so neatly that eliminates the \(xy\) term entirely.
With a fresh set of axes, the equation becomes

• Easier to handle,
• Free from the \(xy\) mix-up.

This transformation is akin to clearing up murky waters in math-centric language!

Conic Sections Standard Form

After efficiently rotating the axes, the conic section equation begins to lose the messy configurations. Converting this rotated equation into 'standard form' is our next ambition. The standard form simplifies into a more recognizable and friendly version depending on the conic section at hand (ellipse, parabola, or hyperbola).
What does standard form really mean? It typically involves aligning your conic equation to the simplest recognizable form, such as a circle, ellipse, parabola, or hyperbola. In practical terms:

• For an ellipse or circle, you'll translate it into \((x - h)^2/a^2 + (y - k)^2/b^2 = 1\)
• For a parabola, your equation should resemble \((x - h)^2 = 4p(y - k)\)
• For a hyperbola it should become \((x - h)^2/a^2 - (y - k)^2/b^2 = 1\)

With crucial simplification steps complete, the quadratic no longer seems daunting. The conic section in the standard form is ready to interpret and draw. Not only does this form provide visual aid for graphing, but it significantly aids comprehension of basic geometric properties.

Graph Transformation

Once the equation is stripped to standard form, it's ripe for sketching its graph. Understanding graph transformation reinforces visual comprehension and mathematical insights.
When plotting this equation, visualize how the rotations and simplifications affect the shape and orientation of the graph. Conic sections take distinct places on the Cartesian plane:

• Circles appear as round figures.
• Ellipses are elongated, oval shapes.
• Parabolas open upward, downward, or sideways, resembling a U-shape.
• Hyperbolas have dual curves, parting sideways or very steeply.

During the graphing, ensure to display both the original and rotated axes. Watching this transformation visually iterates how algebra and geometry collaborate to create solutions. It offers a crucial learning touchstone for truly grasping the connectedness between forms, angles, and transformations in conic sections.