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

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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 \(\text{slope} = \frac{\Delta y}{\Delta x}\).

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.

A vertical line has the same x-coordinate for all points, so the change in x, \(\Delta x\), is zero along this line.

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.

Therefore, the slope of the line \(\theta = \frac{\pi}{2}\) is undefined because it is a vertical line with no horizontal change.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.