Problem 56 - Precalculus with Calculus Previews

Question

Find a rectangular equation that has the same graph as the given polar equation. $$ r^{2} \cos 2 \theta=16 $$

Step-by-Step Solution

Answer

The rectangular equation is \( x^2 - y^2 = 16 \).

Step 1: Understanding Polar Equation

The given polar equation is \( r^2 \cos 2\theta = 16 \). Our goal is to convert this into a rectangular form using relationships between polar and rectangular coordinates.

Step 2: Using Polar to Rectangular Conversion Formulas

Recall the relationships: \( r^2 = x^2 + y^2 \), \( x = r \cos\theta \), and \( y = r \sin\theta \). Also, note that \( \cos 2\theta = \cos^2\theta - \sin^2\theta \). Thus, \( \cos 2\theta = \frac{x^2 - y^2}{x^2 + y^2} \).

Step 3: Substituting in Known Values

Substitute \( r^2 = x^2 + y^2 \) and \( \cos 2\theta = \frac{x^2 - y^2}{x^2 + y^2} \) into the original polar equation: \((x^2 + y^2) \cdot \frac{x^2 - y^2}{x^2 + y^2} = 16 \).

Step 4: Simplifying the Equation

Simplify the equation by noticing that \((x^2 + y^2) \cdot \frac{x^2 - y^2}{x^2 + y^2} = x^2 - y^2 \). Therefore, the equation becomes \( x^2 - y^2 = 16 \).

Step 5: Rectangular Equation

The rectangular equation corresponding to the given polar equation is \( x^2 - y^2 = 16 \). This equation is a hyperbola.

Key Concepts

Polar Coordinates

Polar coordinates are a way of representing points in a plane using a distance and an angle. Unlike the more common Cartesian (rectangular) coordinates, which use a grid of horizontal (x) and vertical (y) axes, polar coordinates use:

• **r**: the radial distance from a fixed point, known as the pole (similar to the origin in Cartesian coordinates).
• **θ (theta)**: the angle measured counterclockwise from the positive x-axis to the line connecting the point to the pole.

This system is especially useful in situations where circular or rotational symmetry is present, such as in trigonometry or when analyzing periodic functions. In polar coordinates, different equations can describe shapes like circles, spirals, or other conic sections, making them very versatile in mathematics. The main challenge when using polar coordinates is converting them into rectangular coordinates when necessary.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are the most familiar way of representing positions in a plane. They use two perpendicular axes:

• **x-axis**: the horizontal component.
• **y-axis**: the vertical component.

Each point in this system is defined by an ordered pair \(x, y\), where \(x\) represents the distance along the x-axis and \(y\) represents the distance along the y-axis. Rectangular coordinates are particularly useful for tasks involving straight lines and right angles, as they make calculations simpler and more intuitive.
The formulas for converting polar coordinates to rectangular coordinates are \(x = r \cos \theta\) and \(y = r \sin \theta\). These conversions allow us to take advantage of the strengths of both coordinate systems, particularly when dealing with equations or graphs that fit more naturally into one format or the other.

Conic Sections

Conic sections are curves obtained by intersecting a plane with a double-napped cone. Depending on the angle and position of the intersection, different shapes can be formed:

• **Circle**: formed when the plane is perpendicular to the cone's axis.
• **Ellipse**: formed when the plane cuts through the cone at an angle, but not steep enough to reach the base on both sides.
• **Parabola**: formed when the plane is parallel to the surface of the cone.
• **Hyperbola**: formed when the plane intersects both naps of the cone.

In polar to rectangular conversion, understanding the type of conic section represented by the equation is crucial for graphing or simplifying equations. For example, the conversion of the polar equation \(r^2 \cos 2\theta = 16\) into the rectangular equation \(x^2 - y^2 = 16\) reveals a hyperbola, indicating that the graph will have two symmetric curves opening along the x-axis. Knowing these forms helps in visualizing the graph and understanding the nature of the equations involved.