Relative Positions of a Line and a Circle
Big Idea: A line and a circle in the same plane can interact in exactly three ways — they can miss each other entirely, just barely touch, or cross through each other. The key to understanding which case occurs is comparing the distance from the center of the circle to the line with the radius of the circle.
Key Definitions
- Circle: The set of all points in a plane that are at a fixed distance $r$ (the radius) from a fixed point $O$ (the center).
- Line: A straight path that extends infinitely in both directions.
- Distance from a point to a line ($d$): The length of the perpendicular segment drawn from the point to the line. If the line has equation $Ax + By + C = 0$ and the center is $O(x_0, y_0)$, then: $$d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}}$$