Problem 14 - Precalculus: Mathematics for Calculus

Question

Determine the equation of the given conic in XY-coordinates when the coordinate axes are rotated through the indicated angle. $$x y=x+y, \quad \phi=\pi / 4$$

Step-by-Step Solution

Answer

The transformed equation is \(X^2 - 2\sqrt{2}X - Y^2 = 0\).

Step 1: Understand the problem

We have a conic section given by the equation \(xy = x + y\) and need to transform it by rotating the coordinate axes by an angle \(\phi = \pi/4\). The goal is to find the transformed equation in the new coordinates.

Step 2: Use rotation formulas

To rotate the coordinates, use the transformations: \(x = X \cos(\phi) - Y \sin(\phi)\) and \(y = X \sin(\phi) + Y \cos(\phi)\). Substitute \(\phi = \pi/4\):- \(x = \frac{1}{\sqrt{2}}(X - Y)\)- \(y = \frac{1}{\sqrt{2}}(X + Y)\)

Step 3: Substitute and simplify

Substitute the formulas into the original equation:\[ \left( \frac{1}{\sqrt{2}} (X - Y) \right) \left( \frac{1}{\sqrt{2}} (X + Y) \right) = \frac{1}{\sqrt{2}} (X - Y) + \frac{1}{\sqrt{2}} (X + Y) \]Simplify each side separately.

Step 4: Simplify left side of the equation

Perform multiplication on the left side:\[ \frac{1}{2} (X - Y)(X + Y) = \frac{1}{2} (X^2 - Y^2) \]

Step 5: Simplify right side of the equation

Add the terms on the right side:\[ \frac{1}{\sqrt{2}}(X - Y) + \frac{1}{\sqrt{2}}(X + Y) = \frac{1}{\sqrt{2}}(2X) = \sqrt{2}X \]

Step 6: Combine and rearrange the equation

Set the simplified equations from both sides to get:\[ \frac{1}{2}(X^2 - Y^2) = \sqrt{2}X \]Multiply through by 2 to clear the fraction:\[ X^2 - Y^2 = 2\sqrt{2}X \]Rewrite as:\[ X^2 - 2\sqrt{2}X - Y^2 = 0 \]

Step 7: Conclusion

The equation of the conic after rotating the coordinate axes by \(\pi/4\) is:\[ X^2 - 2\sqrt{2}X - Y^2 = 0 \]

Key Concepts

Coordinate Rotation

When solving problems that involve conic sections like ellipses, hyperbolas, or parabolas, sometimes we need to change the perspective by rotating the coordinate system. This is known as coordinate rotation. The main idea is to change how we view the shape to simplify the equation or to better understand the conic's orientation.

To rotate the coordinates, one uses the rotation formulas:

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

These formulas help us shift the standard axis angles \((x,y)\) by an angle \( \phi \), often simplifying the conic equation.

In the given problem, the rotation angle is \( \phi = \pi/4 \), or 45 degrees. By substituting \(\pi/4\) into our formulas, we get new transformed coordinates expressed in terms of \(X\) and \(Y\) that replace the original \(x\) and \(y\). This rotation helps us recast the given conic section's equation into a potentially easier-to-analyze form.

Equation Transformation

After identifying the need for a rotation, the next step is transforming the equation using the new coordinates. This is called equation transformation. It involves substituting the rotated coordinate expressions back into the given equation to see how it looks from the new perspective.

In the example \(xy = x + y\), after applying the rotation formulas, the equation becomes:

• Left Side: \( \left( \frac{1}{\sqrt{2}}(X - Y) \right)\left( \frac{1}{\sqrt{2}}(X + Y) \right) = \frac{1}{2}(X^2 - Y^2) \)
• Right Side: \( \frac{1}{\sqrt{2}}(X - Y) + \frac{1}{\sqrt{2}}(X + Y) = \sqrt{2}X \)

Then, combine these results to simplify the entire equation together.

This process of equation transformation allows us to restate the conic in forms that can reveal properties not initially obvious, like symmetry or axis alignment, that are more easily studied or utilized.

Precalculus Mathematics

Understanding and working with conic sections is a key aspect of precalculus mathematics. Conic sections are the curves obtained by intersecting a cone with a plane in different ways. These include circles, ellipses, parabolas, and hyperbolas.

Identifying the properties of these shapes is critical. This helps in solving real-world problems and prepares for advanced calculus concepts. Precalculus involves not only plotting these sections but also transforming them into different forms using algebraic techniques such as completing the square or rotating the axes, as shown in this exercise.

The exercise of finding the transformed equation via coordinate rotation is fundamental. It enhances our understanding of how shifts in perspective can simplify equations and reveal essential properties of these conic shapes. Mastery of such transformations forms a bridge to deeper mathematical concepts and applications.