Problem 44 - 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. $$ 3 x^{2}+10 x y+3 y^{2}+10=0 $$

Step-by-Step Solution

Answer

Rotate axes by 45°, no further translation needed; intersecting lines result.

Step 1: Identify the Coefficients

Let's recognize the coefficients from the quadratic equation. The given equation is: \[ 3x^2 + 10xy + 3y^2 + 10 = 0 \] The coefficients are: \( A = 3 \), \( B = 10 \), \( C = 3 \).

Step 2: Calculate the Angle of Rotation

Use the formula \( \tan 2\theta = \frac{B}{A-C} \) to find the angle of rotation \( \theta \). Here:\[ \tan 2\theta = \frac{10}{3-3} = \infty \] This implies \( 2\theta = \frac{\pi}{2} \) or \( \theta = \frac{\pi}{4} \).

Step 3: Apply the Rotation of Axes

Rotate the axes by substituting \( x = x'\cos(\theta) - y'\sin(\theta) \) and \( y = x'\sin(\theta) + y'\cos(\theta) \), where \( \theta = \frac{\pi}{4} \). Substitute \( \cos \frac{\pi}{4} = \sin \frac{\pi}{4} = \frac{\sqrt{2}}{2} \):\[ x = \frac{\sqrt{2}}{2}(x' - y') \]\[ y = \frac{\sqrt{2}}{2}(x' + y') \]

Step 4: Substitute and Simplify the Equation

Substitute in the rotated equation:\[ 3\left(\frac{\sqrt{2}}{2}(x' - y')\right)^2 + 10\left(\frac{\sqrt{2}}{2}(x' - y')\right)\left(\frac{\sqrt{2}}{2}(x' + y')\right) + 3\left(\frac{\sqrt{2}}{2}(x' + y')\right)^2 + 10 = 0 \]Expanding this, we simplify to: \[ 4x'^2 + 24x'y' + 4y'^2 + 10 = 0 \]

Step 5: Simplify to Standard Form by Translating Axes

Complete the square for \( 24x'y' \). Since the \( x' \) and \( y' \) terms cancel out thanks to symmetry, the necessary translation can be avoided. Thus, the simplified form remains as itself after a rotation.

Step 6: Summarize and Graph the Equation

The equation simplifies to a graph as a pair of lines, since it forms in the form \( (x' + y')^2 = 0 \) in terms of \( x' \) and \( y' \). Therefore, the expression can be visualized as diagonal lines intersecting at \( x' = -5 \) when substituting back to the original orientation of the problem.

Key Concepts

Rotation of Axes

The rotation of axes is a technique used in algebra and geometry to simplify equations and eliminate cross-product terms from conic sections. In the given equation, \(3x^2 + 10xy + 3y^2 + 10 = 0\), the presence of the \(xy\) term indicates the need for rotation.
To achieve this, we rotate the coordinate system by an angle \(\theta\), calculated using the formula \(\tan 2\theta = \frac{B}{A-C}\). In this case, \(A = 3\), \(B = 10\), and \(C = 3\), leading to \(\tan 2\theta = \frac{10}{0} = \infty \).
Thus, \(2\theta = \frac{\pi}{2}\) and \(\theta = \frac{\pi}{4}\). Rotating the axes helps transform the original equation into one without the \(xy\) cross term, making it easier to interpret and solve. By substituting \(x = x'\cos(\theta) - y'\sin(\theta)\) and \(y = x'\sin(\theta) + y'\cos(\theta)\), we rewrite the equation in terms of \(x'\) and \(y'\).
This transformation can also provide insights about symmetry and alignment, offering a clearer perspective on the geometric properties of the shape, particularly when dealing with complex conic sections.

Trigonometric Transformation

Trigonometric transformations play a crucial role in the process of axis rotation. This involves using trigonometric identities to express the original coordinates into a new set of coordinates after a prescribed rotation angle \(\theta\).
In our example, after determining \(\theta = \frac{\pi}{4}\), the coordinate transformation becomes:\

• \(x = \frac{\sqrt{2}}{2}(x' - y')\)
• \(y = \frac{\sqrt{2}}{2}(x' + y')\)

These formulas derive from substituting the value \(\frac{\pi}{4}\) into the trigonometric functions \(\cos(\theta)\) and \(\sin(\theta)\), both equaling \(\frac{\sqrt{2}}{2}\). The transformation rotates the axes counterclockwise by 45 degrees.
This operation helps eliminate cross-product terms in quadratic equations (like the \(xy\) term) by aligning one of the axes with the principal axes of the conic section, simplifying further calculations and visualizations.

Quadratic Equation

The equation we are working with is a quadratic equation with two variables, expressed as \(3x^2 + 10xy + 3y^2 + 10 = 0\). Quadratic equations in two variables generally describe the conic sections: circles, ellipses, hyperbolas, and parabolas.
Recognizing the coefficients \(A\), \(B\), and \(C\) in the general form \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\), allows us to determine the shape and characteristics of the conic described by the equation. In particular, the \(xy\) term affects the orientation, suggesting a rotation of axes is necessary.
Post-rotation, the goal is to reduce the equation into a recognizable form, such as \((x' + y')^2 = 0\), which represents two coincident lines. Completing the square may also be used as a manipulation technique, but in this instance, due to the perfect symmetry formed by the terms after rotation, further operation was unnecessary for simplification.

Coordinate Geometry

Coordinate geometry, or analytic geometry, uses algebraic methods to represent and solve geometric problems using coordinates. It merges algebra and geometry, offering a powerful set of tools for understanding conic sections like the given equation. By plotting equations on a coordinate plane, we gain visual insights into their properties and behaviors.
When handling quadratic equations involving conic sections, coordinate geometry enables the transformation of axes to clarify and simplify equation forms. This means graphically analyzing aspects like intersections, tangents, and other geometric properties.
In the final step of our problem, once the cross-product term is eliminated and reconfiguration complete, the graph depicts diagonal intersection lines at \(x' = -5\) when adjusting back to original coordinates.
This visualization helps consolidate abstract algebraic concepts with concrete geometric interpretations, reinforcing learning through graphical representation.