Problem 22 - 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.) \(3 x^{2}-2 \sqrt{3} x y+y^{2}=1\)

Step-by-Step Solution

Answer

The graph is an ellipse after rotating the axes.

Step 1: Identify the Coefficient of the Cross Product Term

The original equation is given as \(3x^2 - 2\sqrt{3}xy + y^2 = 1\). The coefficient of the cross product term \(xy\) is \(-2\sqrt{3}\). Our goal is to eliminate this term by rotating the axes.

Step 2: Determine the Rotation Angle

To eliminate the \(xy\) term, we need to use the rotation formula \(tan(2\theta) = \frac{B}{A-C}\), where \(A=3\), \(B=-2\sqrt{3}\), and \(C=1\). Substituting the values, we have: \[ \tan(2\theta) = \frac{-2\sqrt{3}}{3-1} = -\sqrt{3} \]The angle \(\theta\) that satisfies \(\tan(2\theta) = -\sqrt{3}\) is \(2\theta = 120^\circ\) or \(\theta=60^\circ\) (or equivalently \(\theta = -30^\circ\)).

Step 3: Compute Rotation Matrix and New Coordinates

The rotation of axes uses the matrix: \[\begin{pmatrix}x' \y'\end{pmatrix} = \begin{pmatrix}\cos\theta & \sin\theta \-\sin\theta & \cos\theta\end{pmatrix}\begin{pmatrix}x \y\end{pmatrix}\]At \(\theta=60^\circ\), \(\cos 60^\circ = \frac{1}{2}\) and \(\sin 60^\circ = \frac{\sqrt{3}}{2}\), substitute these in to find:\[x' = \frac{1}{2}x + \frac{\sqrt{3}}{2}y, \, y' = -\frac{\sqrt{3}}{2}x + \frac{1}{2}y\]

Step 4: Substitute and Eliminate Cross Product Term

Substitute the expressions for \(x\) and \(y\) in terms of \(x'\) and \(y'\) into the original equation. Simplifying the equation should eliminate the cross product term \(xy\). After substitution:\[ 3\left(\frac{1}{2}x' + \frac{\sqrt{3}}{2}y'\right)^2 - 2\sqrt{3}(\frac{1}{2}x' + \frac{\sqrt{3}}{2}y')( -\frac{\sqrt{3}}{2}x' + \frac{1}{2}y') + \left(-\frac{\sqrt{3}}{2}x' + \frac{1}{2}y'\right)^2 = 1\]Simplify to get an equation free of the \(x'y'\) term.

Step 5: Identify the Simplified Equation Type

Simplify the expression. The new equation should now look something like \(Ax'^2 + Cy'^2 = 1\) or a similar form. This equation describes a conic section depending on the coefficients. The simplified version shows\[x'^2 + 4y'^2 = 1\]which represents an ellipse.

Key Concepts

Coordinate Rotation

Coordinate rotation is a technique used in mathematics to transform the plane. It's particularly helpful in simplifying equations by changing the variables. When you rotate the coordinate axes, you apply a rotation matrix to the original coordinates, usually labeled as

• Original coordinates:
  • \( x \)
  • \( y \)
• New coordinates:
  • \( x' \)
  • \( y' \)

The formula to do this involves trigonometric functions based on the rotation angle \( \theta \). Specifically, the relationships are:

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

Applying a rotation allows simplification of equations, such as removing troublesome terms like the cross-product term \(xy\). The correct rotation angle is found by solving \(\tan(2\theta)\), often leading to equations with easier forms.

Conic Sections

Conic sections are curves obtained by intersecting a cone with a plane. The main types of conic sections include:

• Circles
• Ellipses
• Parabolas
• Hyperbolas

Each conic section has a distinct equation form. These equations can be rotated or translated in the coordinate plane.The equation of a conic may initially include mixed or cross-product terms like \(xy\), especially when tilted. By rotating the coordinate system, we can simplify these equations into their more recognizable forms. This is often needed to correctly identify and understand their geometric properties.

Ellipse

An ellipse is a specific type of conic section and can be visualized as an elongated circle. Its standard equation is often expressed as:\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]where:

• \(a\) is the semi-major axis
• \(b\) is the semi-minor axis

If \(a = b\), it forms a circle. Upon rotation, even if the original equation for an ellipse contains a cross-product term \(xy\), properly rotating the coordinates allows us to eliminate this term. This turns the equation into a simpler form without the \(xy\) term, making it easier to recognize the properties of an ellipse. The equation derived from the exercise is \( x'^2 + 4y'^2 = 1 \). This confirms it's an ellipse since it lacks the cross-product term upon rewriting.

Equation Transformation

Equation transformation is a mathematical process that modifies the appearance of an equation without altering its fundamental properties. For conic sections, it involves simplifying the equation to remove complex terms like the cross-product \(xy\). To transform an equation effectively:

• Identify the necessary parameters (like \(A\), \(B\), and \(C\) in the quadratic equation).
• Solve for the optimal rotation angle \(\theta\) using the equation \( \tan(2\theta) = \frac{B}{A-C} \).
• Use trigonometric identities to compute the new coordinates \(x'\) and \(y'\).
• Substitute the new variables back into the original equation to achieve a cross-product-free form.

The removal of the \(xy\) term simplifies the task of recognizing the conic section type from its equation. This allows mathematicians and students to manipulate, understand, and interpret geometric shapes more easily in coordinate geometry.