Circle of Apollonius

Visualize the locus of points with a constant distance ratio to two foci.

Input Parameters

Focus Distance (a)

Distance Ratio (k)

Geometric Context

Focus A: (-5, 0)

Focus B: (5, 0)

Point P(x, y) satisfies: PA / PB = 2

Circle Properties

Circle

Locus Type

Points P such that PA / PB = k

Radius (R)6.67

Center (x₀, 0)8.33

Quick Reference

k = 1Linear Bisector

k > 1Circle around B

k < 1Circle around A

k → 0 or k → ∞Circle collapses to point

What is the Circle of Apollonius?

In geometry, the Circle of Apollonius is defined as the set of all points P whose distances from two fixed points A and B have a constant ratio k. If k = 1, the locus is the perpendicular bisector of the segment AB. If k ≠ 1, the locus is a circle.

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Pro Tips

Foci are assumed to be at (-a, 0) and (a, 0).
When k > 1, the circle encloses the focus at (a, 0).
When k < 1, the circle encloses the focus at (-a, 0).
As k approaches 1, the circle's radius approaches infinity and its center moves further away.

Fun Facts

"Apollonius of Perga (c. 262 – c. 190 BC) was a Greek geometer known as 'The Great Geometer'."
"The Apollonius circle is related to the concept of 'isogonal conjugates' in triangle geometry."
"The Apollonian gasket is a fractal generated from circles tangent to one another."

Formula: Radius R = |2ak / (k² - 1)|, Center x₀ = a(k² + 1) / (k² - 1)

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