Problem 50 - Precalculus: Functions and Graphs

Question

Find an equation in \(x\) and \(y\) that has the same graph as the polar equation. Use it to help sketch the graph in an \(r \theta\) -plane. $$r^{2} \sin 2 \theta=-10$$

Step-by-Step Solution

Answer

The equation is \(2xy = -10\); it represents a hyperbola.

Step 1: Understand the Polar Equation

The given equation is a polar equation where \(r^2 \sin 2\theta = -10\). Here, \(r\) is the radial distance from the origin and \(\theta\) is the angle from the positive x-axis.

Step 2: Convert Polar to Rectangular Coordinates

We know the relationships between polar and rectangular coordinates: \(x = r\cos\theta\), \(y = r\sin\theta\). Also use the identity \(\sin 2\theta = 2\sin\theta\cos\theta\). Substitute these into the equation:\[r^2 \cdot 2 \sin\theta \cos\theta = -10.\]

Step 3: Relate to Rectangular Terms

Since \(r^2 = x^2 + y^2\), and using \(\sin\theta = \frac{y}{r}\) and \(\cos\theta = \frac{x}{r}\), substitute these back:\[(x^2 + y^2) \cdot 2 \left(\frac{y}{r}\right) \left(\frac{x}{r}\right) = -10.\] This simplifies to:\[2xy = -10.\]

Step 4: Simplify the Rectangular Equation

The equation \(2xy = -10\) is already simplified in rectangular coordinates. This equation is a hyperbola in the xy-plane.

Step 5: Analyze the Shape in the Polar Plane

In the polar \(r\theta\)-plane, the original equation \(r^2 \sin 2\theta = -10\) suggests a figure with symmetry about the origin and opening based on the sine component, corresponding to a rotated hyperbola in the rectangular xy-plane.

Key Concepts

Polar Equations

Polar equations are mathematical expressions that specify a relationship between the radius \(r\) and the angle \(\theta\) in the polar coordinate system. In a polar equation, such as \(r^2 \sin 2\theta = -10\), the radius \(r\) indicates how far a point is from the origin, while \(\theta\) represents the angle formed with the positive x-axis.Unlike rectangular (Cartesian) coordinates that use \(x\) and \(y\) coordinates, polar coordinates offer a convenient way to express curves and shapes that exhibit rotational symmetry. They are particularly useful when dealing with spirals, circles, and any graph where the radial distance from a central point is of interest.When dealing with polar equations, remember:

• \(r\) can be positive or negative, affecting the direction of the graph.
• \(\theta\) is usually measured in radians.
• Polar equations can often be simplified or transformed into different coordinate systems.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, use two numbers, \(x\) and \(y\), to describe a point on a plane. These coordinates are based on a grid system with horizontal and vertical axes, making them intuitive and easy for many practical applications.The relationship between polar and rectangular coordinates is pivotal for converting between systems:

• \(x = r \cos\theta\): This relates the horizontal component based on the angle and radial distance.
• \(y = r \sin\theta\): This determines the vertical component similarly.

This conversion allows us to translate polar graphs into the xy-plane, which can be crucial for graphing in rectangular coordinates, where most traditional plotting techniques are used. It bridges the gap between intuitive graphical representation and polar equations.

Hyperbola

A hyperbola is a type of conic section that is represented by an equation and has two disconnected curves. In the rectangular coordinate system, the equation of a hyperbola often takes the form \(xy = c\), where \(c\) is a constant.In the context of our original problem, the polar equation \(r^2 \sin 2\theta = -10\) is converted to a rectangular form resulting in \(2xy = -10\), clearly showcasing a hyperbolic equation. This demonstrates the transformation between coordinate systems and how shapes like hyperbolas manifest differently in polar and rectangular forms.Key features of hyperbolas include:

• Two asymptotes that guide the general trajectory of each branch.
• Distinct branches that open away from each other.
• Symmetry about the center of the hyperbola.

Hyperbolas reflect how distinct orbits and graphs behave through space and can appear rotated when derived from polar equations.

Coordinate System Conversion

The conversion between polar and rectangular coordinates is essential in many areas of mathematics and engineering, especially when dealing with complex curves and shapes.To convert from polar to rectangular coordinates:

• Apply the transformations \(x = r \cos\theta\) and \(y = r \sin\theta\).
• Simplify to express \(r\) and \(\theta\) terms using \(x\) and \(y\).
• Use trigonometric identities to handle equations that include functions like \(\sin 2\theta\).

Coordinate system conversion allows for the application of widely-used mathematical tools and graphing techniques traditionally housed in the rectangular system. It also provides a clearer view of graph behaviors, helping us transition complex polar equations into more manageable forms. This process is crucial in visualizing geometric properties and spatial relations efficiently.