Problem 79 - Precalculus

Question

Find the polar equation that is equivalent to a vertical line, \(x=a\).

Step-by-Step Solution

Answer

The polar equation for a vertical line \( x = a \) is \( r = \frac{a}{\cos \theta} \).

Step 1: Recall the Cartesian Equation for a Vertical Line

The Cartesian equation for a vertical line is given by \( x = a \) where \( a \) is a constant. This line is parallel to the y-axis and intersects the x-axis at the point \( (a, 0) \).

Step 2: Express x in Terms of Polar Coordinates

In polar coordinates, the relationship between Cartesian and polar coordinates is given by \( x = r \cos \theta \). Here, \( r \) is the radius or the distance from the origin, and \( \theta \) is the angle with the positive x-axis.

Step 3: Substitute into the Cartesian Equation

Substitute \( x = r \cos \theta \) into the equation \( x = a \). This gives the equation \( r \cos \theta = a \).

Step 4: Simplify the Polar Equation

The polar equation form of the vertical line \( x = a \) is \( r = \frac{a}{\cos \theta} \). This respects the characteristics of the vertical line in Cartesian coordinates.

Key Concepts

Vertical Line

A vertical line is a straightforward yet crucial concept in geometry. It represents a line that runs up and down the coordinate plane, parallel to the y-axis, and does not slant to the left or right. This means that all points on this line have the same x-coordinate.
In other words, for any point y on a vertical line, the x-value remains consistent.

• For example, consider the vertical line defined by the equation x = a , where a is a constant. Every point on this line will have x = a , such as (a, 0) , (a, 1) , and (a, -1) .

Understanding vertical lines is vital for converting between coordinate systems because they highlight how direction remains consistent relative to an axis.

Cartesian Equation

A Cartesian equation is an equation that represents a relation using Cartesian coordinates (x, y) . It's named after the renowned philosopher and mathematician René Descartes, who developed this coordinate system.
In the Cartesian plane, equations take the form of fixed or variable relationships between x and y . For a vertical line, the equation is written as x = a , where a is a constant that specifies the line's horizontal position.
This reflects that all points on this line have the same x-coordinate, meaning the line is perfectly vertical. Cartesian equations are essential for defining simple lines and curves, allowing for easy graphical interpretations.

Polar Equation

In mathematics, a polar equation expresses a relationship using polar coordinates (r, \theta), where r is the radial distance from the origin, and \(\theta\) is the angle measured from the positive x-axis. In the context of our exercise, the problem is to convert a vertical line from its Cartesian form to a polar form.
When you begin this conversion, the Cartesian equation x = a is replaced using the polar equivalent x = r \cos \theta.Substituting into the equation yields r \cos \theta = a. Upon rearranging, the polar equation becomes r = \frac{a}{\cos \theta}.

• This polar equation respects the geometry of the original vertical line. It helps bridge the gap between Cartesian and polar descriptions, helping us understand more about spatial relationships.

Understanding polar equations allows for a better grasp of curves and lines in a polar coordinate system, expanding our ability to interpret geometrical information.

Polar Coordinates

Polar coordinates provide a unique way of describing the position of a point in a plane. Unlike the Cartesian system—where points are located using x and y coordinates—polar coordinates use a radial distance and an angular direction to define a position.
The system involves two parameters:

• r: the distance from the point to the origin.
• \(\theta\): the angle from the positive x-axis to the line formed by connecting the point to the origin.

This means that any point in a plane can be represented by these two values. Transforming from Cartesian to polar coordinates can reveal different aspects of geometric problems.
In the exercise context, polar coordinates enable, among other things, the conversion of straightforward lines into valuable forms useful for specific applications in physics and engineering.