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 transformed equation represents a hyperbola.

Step 1: Formulate the rotation transformation

To rotate the coordinate system by \(45^{\circ}\), we use the transformation formulas: \( x = X \cos(\theta) - Y \sin(\theta) \) and \( y = X \sin(\theta) + Y \cos(\theta) \). For a \(45^{\circ}\) rotation, \(\cos(45^\circ) = \sin(45^\circ) = \frac{\sqrt{2}}{2}\). Therefore, the equations become: \( x = \frac{\sqrt{2}}{2}(X - Y) \) and \( y = \frac{\sqrt{2}}{2}(X + Y) \).

Step 2: Substitute the new variables

Substitute the rotated coordinates into the original equation: \( \left(\frac{\sqrt{2}}{2}(X - Y)\right)^{1/2} + \left(\frac{\sqrt{2}}{2}(X + Y)\right)^{1/2} = a^{1/2} \). Simplifying inside the roots gives: \( \sqrt{\frac{\sqrt{2}}{2}} (X - Y)^{1/2} + \sqrt{\frac{\sqrt{2}}{2}} (X + Y)^{1/2} = a^{1/2} \).

Step 3: Simplify the expression

Recognize that \( \sqrt{\frac{\sqrt{2}}{2}} = \left(\frac{\sqrt{2}}{2}\right)^{1/4} \). This simplifies the expression to: \( \left(\frac{1}{\sqrt[4]{2}}\right) (X - Y)^{1/2} + \left(\frac{1}{\sqrt[4]{2}}\right) (X + Y)^{1/2} = a^{1/2} \). Factor out \( \frac{1}{\sqrt[4]{2}} \) to get: \( \frac{1}{\sqrt[4]{2}}((X - Y)^{1/2} + (X + Y)^{1/2}) = a^{1/2} \).

Step 4: Eliminate single radical by squaring

Square both sides to remove the first radical: \( \left((X - Y)^{1/2} + (X + Y)^{1/2}\right)^2 = (a^{1/2} \cdot \sqrt[4]{2})^2 \). This simplifies to: \( X - Y + X + Y + 2\sqrt{(X-Y)(X+Y)} = a^{1/2} \cdot 2 \), resulting in \( 2X + 2\sqrt{X^2 - Y^2} = a^{1/2} \cdot 2 \).

Step 5: Square again to remove final radicals

Now, solve for \( \sqrt{X^2 - Y^2} \) and square both sides again to eliminate the remaining radical:\( (a^{1/2} - 2X) = 2\sqrt{X^2 - Y^2} \) becomes \( \left(a^{1/2} - 2X\right)^2 = 4(X^2 - Y^2) \). Expand and simplify: \( (a - 4a^{1/2}X + 4X^2) = 4X^2 - 4Y^2 \).

Step 6: Bring to a standard form

Reorganize the equation to find its relationship: \( a - 4a^{1/2}X = -4Y^2 \) or \( 4Y^2 = -a + 4a^{1/2}X \). Simplifying gives an equation of a hyperbola.

Key Concepts

Rotation of Axes

Rotating the axes is a vital mathematical technique often used to simplify complex equations. When you rotate the axes, you're essentially modifying the coordinate system to obtain a new perspective that might be easier to work with. In the context of this exercise, we rotate the axes by \( 45^{\circ} \). This involves applying transformation formulas:

• \( x = X \cos(\theta) - Y \sin(\theta) \)
• \( y = X \sin(\theta) + Y \cos(\theta) \)

Here, \( \theta \) is \( 45^{\circ} \), making both \( \cos(45^{\circ}) \) and \( \sin(45^{\circ}) \) equal to \( \frac{\sqrt{2}}{2} \). Thus, our transformed equations are:

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

By introducing these new variables, \( X \) and \( Y \), the complexity of the original equation is subtly shifted. Think of it as looking at a tilted picture straight on, allowing for a more natural perspective that can make the math more straightforward to handle.

Elimination of Radicals

Eliminating radicals is a common strategy to simplify equations, transforming them from their complex form into something more tangible. Radicals, like square roots, can often make equations unwieldy. In this exercise, our goal was to remove these radicals to simplify the expression.
Initially, we squared both sides of the equation to ease out the first layer of radicals:
\[\left((X - Y)^{1/2} + (X + Y)^{1/2}\right)^2 = (a^{1/2} \cdot \sqrt[4]{2})^2\]This first operation simplifies the equation but doesn't completely rid us of radicals. We are left with:
\[2X + 2\sqrt{X^2 - Y^2} = a^{1/2} \cdot 2\]To eradicate the remaining radical, we square once more. This additional step gives us:\[\left(a^{1/2} - 2X\right)^2 = 4(X^2 - Y^2)\]Through these logical steps, we effectively eliminate radicals, paving the way for a clearer form of the equation that helps in further mathematical manipulations.

Hyperbola Identification

Identifying the type of conic section represented by an equation is crucial in understanding geometric properties. A hyperbola is a specific type of conic section characterized by its distinct open curves.
In this problem, once the radicals were eliminated, the equation simplified to reveal itself as a hyperbola:
\[4Y^2 = -a + 4a^{1/2}X\]The general equation for a hyperbola is typically seen as:

• \( Ax^2 - By^2 = C \)

Where \( A \) and \( B \) are coefficients, and the signs dictate the type of conic section. Here, recognizing the \( Y^2 \) term becomes vital since having a negative sign indicates a hyperbola when contrasted with \( X^2 \).
By breaking down the transformed and simplified equation, the structure of the equation as a hyperbola becomes apparent. This reveals fascinating aspects of the curve's asymptotic nature and how it opens up along the axes, boosting comprehension of the geometric form it takes.