Problem 34 - Thomas Calculus

Question

Replace the polar equations in Exercises \(23-48\) by equivalent Cartesian equations. Then describe or identify the graph. $$ r^{2} \sin 2 \theta=2 $$

Step-by-Step Solution

Answer

The graph is a hyperbola described by the equation \( xy = 1 \).

Step 1: Convert Polar to Cartesian Coordinates

Recall the polar-to-Cartesian conversions: \( x = r \cos \theta \) and \( y = r \sin \theta \). Moreover, \( \, r^{2} = x^{2} + y^{2} \, \) and \( \, \sin 2 \theta = 2 \sin \theta \cos \theta = \frac{2xy}{r^{2}} \. \) Use these to replace \( r^{2} \sin 2\theta \) with Cartesian forms.

Step 2: Substitute and Simplify

Given the equation \( r^{2} \sin 2\theta = 2 \), substitute: \[ r^{2} \sin 2\theta = r^{2} \left( \frac{2xy}{r^{2}} \right) = 2xy \].Thus, the equation becomes \( 2xy = 2 \).

Step 3: Solve for Cartesian Equation

Divide both sides by 2:\[ xy = 1 \].This simplifies the polar equation to its Cartesian form.

Step 4: Identify the Graph

The equation \( xy = 1 \) represents a hyperbola in Cartesian coordinates. Specifically, it is a rectangular hyperbola that is rotated 45 degrees from the standard orientation.

Key Concepts

Polar coordinates

Polar coordinates are a way to represent points in a plane using two values: the radial distance from the origin, denoted as \( r \), and the angular direction \( \theta \). These coordinates are particularly useful in problems involving circles and spiral patterns because they simplify the mathematics involved.

• The distance \( r \) is measured from a fixed central point known as the pole, similar to how you would measure the radius of a circle.
• The angle \( \theta \) is measured from a fixed direction, usually the positive x-axis, and is typically expressed in radians.

This system is powerful for transformations and helps simplify trigonometric equations.

Cartesian coordinates

Cartesian coordinates, in contrast to polar coordinates, describe points in a plane using a pair of numerical values, \( x \) and \( y \). This is the most common coordinate system and is used extensively in algebra and calculus.

• The \( x \)-coordinate expresses the horizontal distance from the origin.
• The \( y \)-coordinate expresses the vertical distance from the origin.

These coordinates allow you to plot points directly on a grid, making it intuitive to visualize shapes like lines, circles, and parabolas.

Conic sections

Conic sections are curves obtained by the intersection of a plane with a cone. Depending on the angle, different shapes can form, such as circles, ellipses, parabolas, and hyperbolas.

• A circle is a special type of ellipse where the plane is parallel to the base of the cone.
• An ellipse forms when the plane intersects the cone at an angle, but not perpendicular to the base.
• A parabola is formed when the plane is parallel to a line on the cone's surface.
• A hyperbola occurs when the plane cuts through both nappes of the cone.

These shapes have unique properties and equations associated with them that are crucial in various fields, including physics and engineering.

Hyperbola

A hyperbola is a type of conic section that appears as two separate curves or "branches." It arises when a plane intersects both the naps of a double cone.

• A hyperbola can be expressed by the equation \( xy = k \) in the case of a rectangular hyperbola, where \( k \) is a constant.
• This can be rotated 45 degrees to form the equation \( xy = 1 \), representing a standard rectangular hyperbola.

Hyperbolas have the unique property of having two foci, and points on the curve maintain a constant difference in distance to these foci.

Coordinate transformation

Coordinate transformation is the process of converting coordinates from one system to another. This is crucial in solving mathematical problems that are more easily defined in a particular coordinate system.Functions such as \( x = r \cos \theta \) and \( y = r \sin \theta \) are used to convert from polar to Cartesian coordinates, allowing equations and graphs to be interpreted in a different form.

• In the original exercise, polar coordinates \((r, \theta)\) were transformed into Cartesian form \((x, y)\).
• This transformation helps in better visualizing and solving equations, particularly those involving trigonometric functions.

Transformations like these are powerful, helping to bridge different mathematical problems and solutions.