Apollonius Distance Ratio
Move from a distance-ratio circle to its line and point limits.
Free theory articleMove from a distance-ratio circle to its line and point limits.
Concept & interpretation
The Apollonius Distance Ratio module explores the geometric locus of a point P such that its distance to a fixed point A is a constant multiple of its distance to another fixed point B. That is, |P - A| = k|P - B| for some constant k. For points A = (-1, 0) and B = (1, 0), this locus generally forms a circle, but takes on degenerate forms like a line or a point for specific values of k. This classical problem bridges distance formulas and algebraic curves.
Experiment
You can interact with the Apollonius Distance Ratio module to see how the geometric locus shifts based on the chosen ratio k and how the horizontal probe y intersects the curve.
- Set the distance ratio k to 0.5 and observe the circular locus. Note the center and radius.
- Set the probe y to 1 to find horizontal intersections with the circle.
- Change the distance ratio k to 1 to see the locus degenerate into the perpendicular bisector (the y-axis). Note how it has one horizontal intersection for any y.
- Change the distance ratio k to 0 to observe the limit where the locus collapses to the point A.