Problem 58 - Calculus Early Transcendentals: Pearson New International Edition

Question

Transform the equation \(x^{1 / 2}+y^{1 / 2}=a^{1 / 2}\) by a rotation of axes through \(45^{\circ}\) and then square twice to eliminate radicals on variables. Identify the corresponding curve.

Step-by-Step Solution

Answer

The equation corresponds to a degenerate line.

Step 1: Rotation of Axes

To perform a rotation of axes through \(45^{\circ}\), we use the rotation formulas: \(x = \frac{X - Y}{\sqrt{2}}\) and \(y = \frac{X + Y}{\sqrt{2}}\), where \(X\) and \(Y\) represent the rotated axes. We'll substitute these into the original equation.

Step 2: Substitute and Simplify

Substitute the rotation formulas into the equation: \(\left(\frac{X-Y}{\sqrt{2}}\right)^{1/2} + \left(\frac{X+Y}{\sqrt{2}}\right)^{1/2} = a^{1/2}\). Simplifying, we have: \(\frac{X^{1/2} - Y^{1/2}}{\sqrt{2}^{1/2}} + \frac{X^{1/2} + Y^{1/2}}{\sqrt{2}^{1/2}} = a^{1/2}\).

Step 3: Simplify Further

Combine the terms on the left using a common denominator: \(\frac{2X^{1/2}}{2^{1/4}} = a^{1/2}\) which simplifies to \(\sqrt{\sqrt{2}} \cdot X^{1/2} = a^{1/2}\). Multiply both sides by \(\sqrt{\sqrt{2}}\) to isolate \(X^{1/2}\): \(X^{1/2} = \frac{a^{1/2}}{\sqrt{\sqrt{2}}}\).

Step 4: Square the Equation

Square both sides to eliminate the square root: \(X = \left(\frac{a^{1/2}}{\sqrt{\sqrt{2}}}\right)^2\). This gives \(X = \frac{a}{\sqrt{2}}\).

Step 5: Recognize the Curve

The equation \(X = \frac{a}{\sqrt{2}}\) is a vertical line in terms of the new rotated axes. Therefore, the original equation corresponds to a degenerate line.

Key Concepts

Radical Equation Transformation

When dealing with radical equations, the main goal is to simplify by eliminating the radicals. This often involves squaring both sides of the equation. To do this effectively, each side of the equation must be carefully analyzed to avoid introducing extraneous solutions.
In the given problem, the original equation is a type of radical equation because it contains square roots: \(x^{1/2}+y^{1/2}=a^{1/2}\). The strategy to eliminate radicals here involves two main steps:

• First, try to express each radical term separately and then square each of them individually. This simplifies the process when radicals are on both sides of the equation.
• Second, if direct simplification isn't possible, consider using a coordinate transformation, such as a rotation, which can often reduce the complexity of the radicals, making the squaring step more straightforward.

Once the radicals are eliminated, as shown in the solution, the resulting equation \(X = \frac{a}{\sqrt{2}}\) becomes algebraically easier to analyze.

Coordinate Transformation

A coordinate transformation is a mathematical operation used to change the position or orientation of a set of points without altering their relative structure. In coordinate geometry, these transformations help simplify complex equations or geometrical problems.
For this exercise, the transformation of interest is the rotation of axes.

• The formula for a rotation by \(45^{\circ}\) is: \(x = \frac{X - Y}{\sqrt{2}}\) and \(y = \frac{X + Y}{\sqrt{2}}\). Here, \((X, Y)\) are the rotated coordinates.
• By substituting these into the original equation, each term is redefined in terms of the new axes. This often simplifies the manipulation of the equation, particularly when symmetric properties can be exploited.

In the current problem, this rotation effectively repositions the equation to allow for easier simplification and identification of the equation's geometric representation.

Analytic Geometry

Analytic geometry, also known as coordinate geometry, involves using algebra to describe and analyze geometric shapes and their properties. This branch of mathematics allows for precise formulation and solution of geometrical problems using equations and coordinates.

• The transformation and manipulation of the given equation are prime examples of analytic geometry in action. By using a rotation of axes, the equation becomes easier to interpret as it relates to simple geometric figures.
• In particular, once simplified, the outcome of the original problem, \(X = \frac{a}{\sqrt{2}}\), is recognized in the form of a line. Specifically, a vertical line in the rotated coordinate system corresponds to a simple linear geometrical entity.
• Recognizing this transformation is crucial in analytic geometry, where the aim is often to understand and describe algebraic equations with specific geometric interpretations.

Through the application of these principles, even complex-looking equations can reveal straightforward and simple geometric shapes.