Euclid defined a circle as the locus of points equidistant...

Euclid defined a circle as the locus of points equidistant from a given point. Apollonius, on the other hand had an alternate definition: Given two points, $\mathrm{A}$ and $\mathrm{B}$, and a constant $\mathrm{k} \neq 1$, the set of all points P such that $\mathrm{PA}=\mathrm{k} \cdot \mathrm{PB}$ is a circle. Consider points $\mathrm{A}(0,0)$ and $\mathrm{B}(b, 0)$, and the constant $\mathrm{k}$. Show that the Apollonian definition does indeed lead to the equation of the circle. Also, find the coordinates of the center and the radius of the circle.

Key Concepts

Apollonian Circle

An Apollonian circle is defined as the set of points in the plane for which the ratio of distances to two fixed points is a constant not equal to one. This characterization generalizes the euclidean definition of a circle and demonstrates that even a non?equi-distant condition (when expressed as a fixed ratio) leads to a circular locus, providing insight into how different definitions can describe the same geometric object.

Distance Formula

In analytic geometry, the distance formula is used to calculate the distance between two points in the plane based on their coordinates. It is essential in transforming geometric definitions into algebraic equations that represent curves, such as rewriting the condition PA = k·PB in squared form to eliminate radicals and facilitate manipulation of the equation.

Equation of a Circle

The standard form of the equation of a circle is written as (x - h)² + (y - k)² = r², where (h, k) represents the center and r the radius. This form is particularly useful when deriving the circle from its definition because it offers a clear geometric interpretation that connects algebraic manipulation with geometric properties.

Completing the Square

Completing the square is an algebraic technique used to transform a quadratic equation into a perfect square trinomial. This method is pivotal in converting the general quadratic equation obtained from the Apollonian condition into the standard form of a circle, thereby revealing the coordinates of the center and the measure of the radius.

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