Problem 43 - Calculus Early Transcendentals: Pearson New International Edition

Question

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. $$ x^{2}+x y+y^{2}=6 $$

Step-by-Step Solution

Answer

The equation reduces to a circle: \( x'^2 + y'^2 = 12 \) with no cross-product term.

Step 1: Identify the Cross-Product Term

The given equation is \( x^2 + xy + y^2 = 6 \). The cross-product term here is \( xy \). Our goal is to eliminate this term by rotating the axes.

Step 2: Formula for Angle of Rotation

To eliminate the \( xy \) term, we use the formula for the angle \( \theta \) of rotation: \( \tan(2\theta) = \frac{B}{A-C} \), where the original equation is \( Ax^2 + Bxy + Cy^2 \). Here, \( A = 1 \), \( B = 1 \), \( C = 1 \).

Step 3: Calculate the Angle of Rotation

Substitute the values into the formula: \( \tan(2\theta) = \frac{1}{0} \). Since the denominator is zero, \( 2\theta = \frac{\pi}{2} \), giving \( \theta = \frac{\pi}{4} \). This means we rotate the axes by 45 degrees.

Step 4: Rotate the Axes

After rotation: \( x = x'\cos(\theta) - y'\sin(\theta) \) and \( y = x'\sin(\theta) + y'\cos(\theta) \). Substitute \( \theta = \frac{\pi}{4} \) to get: \( x = \frac{x'-y'}{\sqrt{2}} \) and \( y = \frac{x'+y'}{\sqrt{2}} \).

Step 5: Substitute New Coordinates into Original Equation

Substitute the rotated coordinates into the original equation: \( (\frac{x'-y'}{\sqrt{2}})^2 + (\frac{x'-y'}{\sqrt{2}})(\frac{x'+y'}{\sqrt{2}}) + (\frac{x'+y'}{\sqrt{2}})^2 = 6 \). Simplify this to \( x'^2 + y'^2 = 12 \).

Step 6: Standard Form of the Equation

The simplified equation \( x'^2 + y'^2 = 12 \) is in the standard form of a circle centered at the origin with radius \( \sqrt{12} = 2\sqrt{3} \).

Step 7: Graph the Equation

Plot the circle with a radius of \( 2\sqrt{3} \) centered at the origin. The original axes were rotated by 45 degrees, so the new axes make a 45-degree angle with the old axes.

Key Concepts

Rotation of Axes

The rotation of axes is a powerful technique in coordinate geometry. It helps simplify equations by eliminating cross-product terms like \(xy\). This makes it easier to analyze and graph the function.
When we rotate axes, we change our perspective on the coordinate plane. This involves introducing new variables, often denoted as \(x'\) and \(y'\), to shift the equation into a form without the interaction term. By applying the correct rotation, the difficult part of the equation disappears.

• The formula \(\tan(2\theta) = \frac{B}{A-C}\) determines the angle of rotation needed to eliminate terms such as \(xy\).
• Substitute values into this formula to find the start of your rotation journey.

So, in the case of the equation \(x^2 + xy + y^2 = 6\), the \(xy\) term disappears with a 45-degree rotation.

Equation of a Circle

The equation of a circle is a familiar concept in geometry, representing points at a constant distance from a center point. Its standard form: \[x'^2 + y'^2 = r^2\]shows a circle centered at the origin with radius \(r\). In our exercise, after rotating the axes and simplifying, we find the circle with equation \(x'^2 + y'^2 = 12\).

• This equation reveals a circle centered at the origin \((0, 0)\).
• The radius of the circle is \(2\sqrt{3}\), which is the square root of 12.

Once simplified and rotated, identifying the circle's equation becomes much more intuitive.

Angle of Rotation

The angle of rotation is crucial when transforming equations. It is the angle through which we rotate the original coordinate axes to achieve our desired transformation. This angle can often be found using the rotation formula.

• Consider \(\tan(2\theta) = \frac{B}{A-C}\) to determine the correct rotation angle.
• In our case, because \(B = 1\), \(A = 1\), and \(C = 1\), we find that \(\theta = \frac{\pi}{4}\) or 45 degrees, due to the denominator becoming zero.

This simulates a perfect rotation, allowing the removal of the cross-term \(xy\) and simplifying the equation's structure.

Coordinate Geometry

Coordinate geometry links algebra and geometry via points on a plane. It uses coordinate systems to describe geometric figures and their properties. The exercise demonstrates how sophisticated transformations can turn complex equations into manageable ones.
In this scenario, we deal with a 2D Cartesian plane, where transformations like rotations clarify our understanding. The endpoints are usually simplified equations or graphs after a transformation.

• Firstly, identify any "cross-product" terms that complicate the equation.
• Apply the rotation of axes technique to eliminate these terms.
• Solve for standard forms to ease plotting on graphs.

This approach refines analysis and visualization within coordinate systems, enhancing problem-solving skills in geometry.