Problem 62 - Calculus Early Transcendentals: Pearson New International Edition

Question

Recall that \(A x^{2}+B x y+C y^{2}+D x+E y+F=0\) transforms to \(a u^{2}+b u v+c v^{2}+d u+e v+f=0\) under a rotation of axes. Find formulas for \(a\) and \(c\), and show that \(a+c=A+C\).

Step-by-Step Solution

Answer

The formulas for "a" and "c" show that \(a + c = A + C\) after considering a rotation of axes.

Step 1: Angle Selection

Choose an angle \(\theta\) for the rotation of axes. This angle determines how the coordinates will be transformed.

Step 2: Coordinate Transformation Equations

Use the coordinate transformation: \(u = x \cos \theta + y \sin \theta\) and \(v = -x \sin \theta + y \cos \theta\). These equations will help in transforming the coordinates from \((x, y)\) to \((u, v)\).

Step 3: Substitute and Expand

Substitute the coordinate transformation equations into the original conic equation \(A x^{2} + B x y + C y^{2} + D x + E y + F = 0\). Then, expand the terms to express everything in terms of \(u\) and \(v\).

Step 4: Identify Coefficients for New Form

Identify the coefficients of terms in the form \(a u^2 + b uv + c v^2 + d u + e v + f = 0\) by comparing the expanded equation with each term separately. This step involves collecting like terms.

Step 5: Formula Derivation for "a" and "c"

Use the trigonometric identities and expanded terms to derive the formulas: \[a = A \cos^2 \theta + B \cos \theta \sin \theta + C \sin^2 \theta \]\[c = A \sin^2 \theta - B \cos \theta \sin \theta + C \cos^2 \theta \]

Step 6: Verification of "a + c"

To verify that \(a + c = A + C\), add the derived formulas for \(a\) and \(c\):\[a + c = (A \cos^2 \theta + B \cos \theta \sin \theta + C \sin^2 \theta) + (A \sin^2 \theta - B \cos \theta \sin \theta + C \cos^2 \theta)\]Notice that the terms involving \(\theta\) cancel each other, leading to \(a + c = A + C\).

Key Concepts

Rotation of Axes

When we talk about the rotation of axes, we refer to changing the angle at which we view the coordinate system, essentially pivoting our perspective around the origin. This mathematical technique is helpful for simplifying complex equations, particularly conic sections. To rotate the axes, we select an angle, \(\theta\), which will determine the degree of this pivot. Such transformations are especially effective for converting oblique conics to standard positions that are much easier to analyze. This is done by altering the coordinate system rather than the object itself, allowing for a simpler new set of equations.Using rotation of axes eliminates cross-product terms like \(xy\), making it easier to identify and classify the conic section you are dealing with.

Coordinate Transformation

Coordinate transformation involves converting points or equations from one system into another. In the context of rotating axes, this transformation is crucial for simplifying the original equation.

• We use transformation equations: \(u = x \cos \theta + y \sin \theta\) and \(v = -x \sin \theta + y \cos \theta\).
• These equations re-map each point from the \((x, y)\) plane to a new \((u, v)\) plane, determined by the rotation angle \(\theta\).

Understanding this concept helps in seeing how complex systems like conics can be simplified through the appropriate use of transformations.

Trigonometric Identities

Trigonometric identities are mathematical relationships between trigonometric functions like sine and cosine. These identities are pivotal in the process of axis rotation and the subsequent transformations.During axis rotation, we use identities such as:

• \(\cos^2 \theta + \sin^2 \theta = 1\)
• \(2\cos \theta \sin \theta = \sin 2\theta\)
• \(\cos 2\theta = \cos^2 \theta - \sin^2 \theta\)

Applying these identities allows us to simplify the expression of \(a\) and \(c\) and subsequently verify the relationship \(a + c = A + C\). Recognizing when and how to use these identities is crucial for manipulating and simplifying transformations effectively.