Question

In Problems \(43-48\), eliminate the cross-product term by a suitable rotation of axes and then, if necessary, translate axes (complete the squares) to put the equation in standard form. Finally, graph the equation showing the rotated axes. x^{2}+x y+y^{2}=6

Short answer

Rotate axes by \( \theta = \frac{\pi}{4} \) to transform the equation to \( x'^2 + y'^2 = 6 \), a circle.

Step-by-step solution

Identify the given quadratic equation

The given equation is \( x^2 + xy + y^2 = 6 \). This is a quadratic equation in two variables with a cross-product term \( xy \). Our goal is to eliminate the cross-product term by rotating the axes.

Calculate the rotation angle

For a general quadratic equation \( Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \), the rotation angle \( \theta \) to eliminate the cross-product term can be found using \( \tan(2\theta) = \frac{B}{A-C} \). Here, \( A = 1, B = 1, C = 1 \), so \( \tan(2\theta) = \frac{1}{1-1} = \infty \), which implies \( 2\theta = \frac{\pi}{2} \) or \( \theta = \frac{\pi}{4} \).

Rotate the axes

The new coordinates \( x', y' \) are obtained using the rotation formulas: \( x = x' \cos\theta - y' \sin\theta \) and \( y = x' \sin\theta + y' \cos\theta \). For \( \theta = \frac{\pi}{4} \), these equations become \( x = \frac{x' - y'}{\sqrt{2}} \) and \( y = \frac{x' + y'}{\sqrt{2}} \).

Transform the quadratic without the cross-product

Substitute \( x = \frac{x' - y'}{\sqrt{2}} \) and \( y = \frac{x' + y'}{\sqrt{2}} \) into the original equation \( x^2 + xy + y^2 = 6 \). After substitutions and simplification, the equation in terms of \( x', y' \) becomes \( x'^2 + y'^2 = 6 \). There is no cross-product term.

Graph the equation in standard form

The equation \( x'^2 + y'^2 = 6 \) is a circle with center at (0,0) and radius \( \sqrt{6} \). Since the equation has no linear terms, no further translation is needed, and the circle can be graphed using the rotated axes.

Complete the process

No additional translation is required because the equation is already in a standard form centered at the origin. Draw the circle centered at (0,0) with radius \( \sqrt{6} \), and indicate the rotated axes that are at a 45-degree angle to the original axes.

Key concepts

Quadratic Equation

A quadratic equation in two variables can be expressed generally as \( Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \). It represents a conic section like a circle, ellipse, parabola, or hyperbola. An important feature of quadratic equations with two variables is the potential presence of a cross-product term \( Bxy \). This term alters the basic shape and orientation of the conic section represented by the equation.

Quadratic equations not having the cross-product term usually represent conics aligned with the coordinate axes, making them easy to visualize and graph. When the cross-product term is present, it suggests a conic that is rotated with respect to the original axes, which could make solving and graphing more challenging. To simplify this, a rotation of axes can be performed to eliminate the term and restore symmetry to the equation.

Eliminate Cross-Product

The elimination of the cross-product term \( xy \) in a quadratic equation requires performing a rotation of axes. This process realigns the axes to make the equation simpler, leading to symmetrical forms like circles or axis-aligned ellipses.

To find the angle of rotation, use the formula \( \tan(2\theta) = \frac{B}{A-C} \), where \( A \), \( B \), and \( C \) are coefficients from the quadratic equation \( Ax^2 + Bxy + Cy^2 + ... = 0 \). For the given problem, \( \tan(2\theta) = \frac{1}{0} = \infty \), indicating that \( \theta = \frac{\pi}{4} \) (45 degrees).

By rotating the axes by \( 45 \) degrees, the cross-product \( xy \) can be eliminated, simplifying the equation to forms that are easier to interpret and graph.

Coordinate Transformation

Coordinate transformation involves applying equations to shift from one set of axes to another, typically by rotating the axes. For the given quadratic equation, the ax rotation is performed using the formulas: \( x = x' \cos \theta - y' \sin \theta \) and \( y = x' \sin \theta + y' \cos \theta \). By substituting \( \theta = \frac{\pi}{4} \), these transformations become \( x = \frac{x' - y'}{\sqrt{2}} \) and \( y = \frac{x' + y'}{\sqrt{2}} \).

After applying this transformation, you can rewrite the entire quadratic equation in terms of the new coordinates \( x' \) and \( y' \), which typically results in the original quadratic terms without the cross-product. This significantly simplifies graphing or analyzing the shape and orientation of any conics represented by the equation.

Graphing Equations

Graphing the transformed quadratic equation helps visually interpret its shape. In the exercise, after switching to the new coordinates, the equation simplifies to \( x'^2 + y'^2 = 6 \). This represents a circle with a radius of \( \sqrt{6} \) centered at the origin (0,0). The rotated axes indicate a 45-degree shift from the original.

When graphing, ensure that:

• The circle is centered at the origin of the new axes.
• The radius of the circle is correctly measured from the center, confirming \( \sqrt{6} \).
• The orientation of the graph shows the rotated axes at a 45-degree angle from the horizontal and vertical.

Visualizing on the right set of axes provides more insight into the geometric properties and symmetries of quadratic equations.