Question

Show that the graph of the equation \(r \cos \theta=a\) is a vertical line \(a\) units to the right of the pole if \(a \geq 0\) and \(|a|\) units to the left of the pole if \(a<0\)

Short answer

The graph of \(r \cos \theta = a\) is a vertical line at \(x = a\), positioned \(a\) units right of the pole if \(a\) is non-negative and \|a|\ units left if \(a\) is negative.

Step-by-step solution

- Understand the Equation

The given equation is in polar coordinates, where \(r\) is the radial distance and \(\theta\) is the angular coordinate. The equation given is \(r \cos \theta = a\).

- Convert to Cartesian Coordinates

To better understand the equation, convert it to Cartesian coordinates. Use the relationships \(x = r \cos \theta\) and \(y = r \sin \theta\). Therefore, \(r \cos \theta = x\). So the equation \(r \cos \theta = a\) becomes \(x = a\).

- Analyze the Cartesian Equation

The equation \(x = a\) represents a vertical line in the Cartesian plane. If \(a \geq 0\), this line is \(a\) units to the right of the y-axis (the pole in polar coordinates). If \(a < 0\), the line is \|a|\ units to the left of the y-axis.

- Conclusion

By comparing the transformed Cartesian equation with the initial polar equation, it is evident that the graph of the equation \(r \cos \theta = a\) is indeed a vertical line. The line's position relative to the pole (origin) depends on the value of \(a\).

Key concepts

Cartesian Coordinates

Cartesian coordinates are used to define a point in a plane using two numerical values. These values are typically represented as \(x\) and \(y\) coordinates.

• The \(x\) coordinate indicates how far along the point is on the horizontal axis.
• The \(y\) coordinate shows the point's position on the vertical axis.

The Cartesian coordinate system is very useful for analyzing geometric shapes and equations.
In this context, the equation \(r \cos \theta = a\) in polar coordinates transforms to \((x = a)\) in Cartesian coordinates.
This makes it easier to understand because \(x = a\) is a straightforward vertical line in the Cartesian plane.

Vertical Line

A vertical line in the Cartesian coordinate system is a line that goes straight up and down.
In mathematical terms, a vertical line has an undefined slope and can be represented by the equation \(x = k\), where \(k\) is a constant.
In the solution, the equation \(r \cos \theta = a\) translates into \(x = a\).
This confirms that the equation represents a vertical line.
If \(a \geq 0\), the line is \(a\) units to the right of the origin.
If \(a < 0\), the line is \(|a|\) units to the left of the origin.
Vertical lines are simple but powerful tools for understanding complex equations when converted into Cartesian form.

Radial Distance

Radial distance is a concept used in polar coordinate systems. It denotes the distance from the origin (called the pole) to a specific point in the plane. It is represented by \(r\).
In the given exercise, the equation \(r \cos \theta = a\) involves \(r\) as the radial distance.
This distance can be different for each point, but for our specific case, it plays a secondary role.
The main insight comes from understanding how \(r \cos \theta\) converts to Cartesian coordinates, simplifying the equation to \(x = a\).
Despite its secondary role in this exercise, radial distance is an essential concept for various applications in polar coordinates and helps in transforming equations to more familiar forms.

Angular Coordinate

In polar coordinates, the angular coordinate (denoted as \(\theta\)) specifies the direction of the point from the pole. It measures the angle from the positive x-axis (or polar axis) to the line connecting the origin to the point.
The angle \(\theta\) is typically measured in radians or degrees.
In the equation \(r \cos \theta = a\), \(\theta\) represents the direction but is indirectly used in this exercise.
When converting to Cartesian coordinates, \(r \cos \theta\) turns into the x-coordinate. Hence, we get a simple vertical line \(x = a\).
Although \(\theta\) doesn't appear in the Cartesian form directly, it is crucial for understanding transformations between coordinate systems and analyzing more complex curves.