Question

Convert the rectangular equation to polar form and sketch its graph. $$ x=8 $$

Short answer

The polar form is \( r = 8 \sec \theta \).

Step-by-step solution

Understanding Rectangular Equation

The given rectangular equation is \( x = 8 \). In rectangular coordinates, this represents a vertical line that is 8 units to the right of the y-axis.

Converting to Polar Coordinates

In polar coordinates, the relationship between rectangular and polar coordinates is given by \( x = r \cos \theta \) and \( y = r \sin \theta \). For the equation \( x = 8 \), apply the transformation: \( 8 = r \cos \theta \). This gives the polar equation \( r \cos \theta = 8 \).

Simplification

Given \( r \cos \theta = 8 \), solve for \( r \) getting \( r = \frac{8}{\cos \theta} = 8 \sec \theta \). This represents the same vertical line in polar coordinates.

Sketching the Graph

To sketch the graph, understand that \( x = 8 \) is a vertical line at 8 units on the x-axis. In polar coordinates, the equation \( r = 8 \sec \theta \) means that for any angle \( \theta \), \( r \) will adjust so the x-coordinate remains at 8, drawing that vertical line when transformed back to rectangular.

Key concepts

Rectangular to Polar Conversion

The process of converting a rectangular equation into polar coordinates involves using specific mathematical relationships. This process allows us to describe geometric shapes or paths in an alternative form. By understanding both coordinate systems, we can interchangeably describe positions or graphs.

Rectangular coordinates describe a point on the plane using an ordered pair \(x, y\). These are the typical Cartesian coordinates. Polar coordinates, however, express a point using the distance from a reference point (the origin) and an angle from a reference direction, typically the positive x-axis. In polar coordinates, a point is defined as \(r, \theta\).

The key equations for converting from rectangular to polar are:

• \(x = r\cos\theta\)
• \(y = r\sin\theta\)
• \(r = \sqrt{x^2 + y^2}\)
• \(\theta = \tan^{-1}\left(\frac{y}{x}\right)\)

To convert the equation \(x = 8\) to polar form, we replace x with \(r\cos\theta\). The result is \(r\cos\theta = 8\). Solving for \(r\) gives \(r = \frac{8}{\cos\theta} = 8\sec\theta\). This beautifully reinterprets the straight line \(x = 8\) within the polar framework.

Graphing Polar Equations

Graphing polar equations involves understanding how changes in \(r\) and \(\theta\) affect the position of a point in the plane. While rectangular graphs are often sketched using x and y coordinates, polar graphs utilize these radial and angular coordinates to plot curves.

In the rectangular equation \(x = 8\), the graph is simply a vertical line crossing the x-axis at 8. However, when graphed in polar coordinates, it takes a unique representation. The polar equation we derived \(r = 8\sec\theta\) indicates that as \(\theta\) varies, \(r\) adjusts to ensure the x-coordinate remains 8, regardless of the angle.

This creates the same vertical line at \(x = 8\), but expressed in terms of polar variables. When sketching this:

• Plot various angles \(\theta\), observing where the line crosses the x-axis.
• Keep the x-component constant due to the nature of secant.
• Recognize that despite its unique polar construction, it represents a standard vertical line.

This helps visualize how a simple linear structure in Cartesian coordinates can translate distinctly within polar systems.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are foundational in mathematics for specifying points on a plane. Through variables \(x\) and \(y\), they build the framework for many fundamental equations and graphs used in algebra and calculus studies.

The \(x\)-axis and \(y\)-axis divide the plane into four quadrants, with each point defined by its relative position compared to these axes. For example, the equation \(x = 8\) describes a line specifically and simply as always having an x-value of 8, regardless of the y-value.

This geometrically means:

• The line runs parallel to the \(y\)-axis.
• This horizontal structure is easy to locate, as it remains fixed in space, never deviating horizontally.
• It is a straight vertical path through the set point of (8, y) where y extends infinitely in both directions.

Rectangular equations such as this demonstrate the clear and straightforward nature of Cartesian plotting systems, acting as a reference point for understanding more complex coordinate transformations like those into polar.