Explain why the slope of the line θ=π/2 is undefined.

Verified step by step guidance

1. Recall that the slope of a line in the Cartesian plane is given by the ratio of the change in y to the change in x, which is \(\text{slope} = \frac{\Delta y}{\Delta x}\).
2. The line \(\theta = \frac{\pi}{2}\) in polar coordinates corresponds to all points where the angle from the positive x-axis is \(90^\circ\), which means the line is vertical.
3. A vertical line has the same x-coordinate for all points, so the change in x, \(\Delta x\), is zero along this line.
4. Since the slope formula involves division by \(\Delta x\), and here \(\Delta x = 0\), the slope expression becomes \(\frac{\Delta y}{0}\), which is undefined because division by zero is not defined in mathematics.
5. Therefore, the slope of the line \(\theta = \frac{\pi}{2}\) is undefined because it is a vertical line with no horizontal change.

Key Concepts

Slope of a Line

The slope of a line measures its steepness and is calculated as the ratio of the change in y to the change in x (rise over run). It indicates how much y changes for a unit change in x.

Vertical Lines and Undefined Slope

Vertical lines have no horizontal change (Δx = 0), making the slope calculation involve division by zero, which is undefined. Therefore, vertical lines do not have a defined slope.

Angle θ = π/2 in Polar Coordinates

The line θ = π/2 in polar coordinates corresponds to the vertical line x = 0 in Cartesian coordinates. Since it is vertical, its slope is undefined due to zero horizontal displacement.