Problem 23 - Thomas Calculus

Question

Rotate the coordinate axes to change the given equation into an equation that has no cross product \((x y)\) term. Then identify the graph of the equation. (The new equations will vary with the size and direction of the rotation you use.) \(\sqrt{2} x^{2}+2 \sqrt{2} x y+\sqrt{2} y^{2}-8 x+8 y=0\)

Step-by-Step Solution

Answer

It's a circle centered at \((4\sqrt{2}, -4\sqrt{2})\) with radius \(\sqrt{32}\).

Step 1: Identify Cross Term

First, observe the given equation \(\sqrt{2} x^{2} + 2\sqrt{2} xy + \sqrt{2} y^{2} - 8x + 8y = 0\). The cross term here is \(2\sqrt{2}xy\). This indicates that the equation needs a rotation of the axes to eliminate this term.

Step 2: Calculate Rotation Angle

To eliminate the cross term \(Bxy\) where \(B = 2\sqrt{2}\), use the formula \(\tan(2\theta) = \frac{B}{A-C}\). Here, \(A = \sqrt{2}\) and \(C = \sqrt{2}\), so \(\tan(2\theta) = \frac{2\sqrt{2}}{0}\). Since \(A=C\), \(\theta = 45^{\circ}\).

Step 3: Perform Rotation of Axes

Apply the rotation transformation using \(x = X\cos(\theta) - Y\sin(\theta)\) and \(y = X\sin(\theta) + Y\cos(\theta)\). Given \(\theta = 45^{\circ}\), substitute \(\cos(45^{\circ}) = \sin(45^{\circ}) = \frac{\sqrt{2}}{2}\). Substitute these into the original equation, replace \(x\) and \(y\) with the expressions in terms of \(X\) and \(Y\).

Step 4: Simplify New Equation

On substituting and simplifying, you will find that the cross term \(XY\) will be eliminated. The equation \(X^2 - 8\sqrt{2}X + Y^2 + 8\sqrt{2}Y = 0\) will be achieved. No outside (\(XY\)) term will be present.

Step 5: Analyze the Result

The simplified equation form \(X^2 - 8\sqrt{2}X + Y^2 + 8\sqrt{2}Y = 0\) represents a circle after completing the square for \(X\) and \(Y\).

Step 6: Identify the Graph

Completing the square for both \(X\) and \(Y\) gives \((X - 4\sqrt{2})^2 + (Y + 4\sqrt{2})^2 = 32\). This is the equation of a circle centered at \((4\sqrt{2}, -4\sqrt{2})\) with radius \(\sqrt{32}\).

Key Concepts

Cross Product Term

In the study of coordinate rotations, one key objective often involves eliminating the cross product term in a given equation. The term itself, such as \(2\sqrt{2}xy\) found in our given equation, can complicate the identification and classification of the graph.
To simplify matters and analyze the geometry more directly, it's essential to remove this term. This is generally achieved through a rotation of the coordinate axes.
Cross product terms, like \(xy\), suggest an inclined symmetry. This means instead of the graph being aligned with the traditional coordinate axes, it might be tilted at a certain angle.
By removing the cross term, we aim to align the graph with our axes, allowing for easier interpretation and identification.

Graph Identification

Graph identification, particularly for conic sections like circles, ellipses, and hyperbolas, requires recognizing the specific equation form.
Once the cross product term is eliminated, the rewritten equation can be easily matched to a standard conic section form.

• For our equation, simplifying it post-rotation resulted in a form closely resembling that of a circle.
• Upon further simplification, involving completing the square for both variables helps to determine the center and radius of the circle.

Understanding how various features of the equation relate to the graph's geometry — like the center from completing the square — is crucial as it brings clarity to what the graph represents.
This may initially seem complex, but removing the cross term significantly aids this identification process.

Rotation Angle Calculation

Determining the correct rotation angle is central to eliminating the cross product term and simplifying the equation.
The angle of rotation, \(\theta\), can be found using the tangent of double the angle: \[ \tan(2\theta) = \frac{B}{A-C} \] where \(B\) is the coefficient of the \(xy\) term, and \(A\) and \(C\) are the coefficients of \(x^2\) and \(y^2\) respectively.

• In this problem, since \(A = C = \sqrt{2}\), the formula simplifies due to division by zero, indicating an easy calculation for a specific case with \(A = C\).
• The result, \(\theta = 45^{\circ}\), was found as it perfectly aligns the graph along the axes after rotation.

Recognizing scenarios where these coefficients exhibit symmetry, as here, can quickly lead to determining the necessary angle for transformation.

Circle Equation

Once the cross product term is removed and the equation simplified, it becomes necessary to rewrite it into the familiar form of a circle equation.
The standard form is \[ (X-h)^2 + (Y-k)^2 = r^2 \] where \((h, k)\) represents the center of the circle and \(r\) its radius.

• After performing a coordinate rotation and simplifying the equation, we completed the square to arrive at \((X - 4\sqrt{2})^2 + (Y + 4\sqrt{2})^2 = 32\).
• From this, the center was identified as \( (4\sqrt{2}, -4\sqrt{2}) \) and the radius as \( \sqrt{32} \).

This conversion into the circle's standard form not only confirms the graph's type but also precisely identifies its spatial characteristics.
Mastering the transition from a general equation to this form makes understanding geometric shapes much more approachable.