Editing Locus (mathematics) (section)

===First example===
Find the locus of a point ''P'' that has a given ratio of distances ''k'' = ''d''<sub>1</sub>/''d''<sub>2</sub> to two given points.

In this example ''k'' = 3, ''A''(−1, 0) and ''B''(0, 2) are chosen as the fixed points.

: ''P''(''x'', ''y'') is a point of the locus
: <math>\Leftrightarrow |PA| = 3 |PB| </math>
: <math>\Leftrightarrow |PA|^2 = 9 |PB|^2 </math>
: <math>\Leftrightarrow (x + 1)^2 + (y - 0)^2 = 9(x - 0)^2 + 9(y - 2)^2 </math>
: <math>\Leftrightarrow 8(x^2 + y^2) - 2x - 36y + 35 = 0 </math>
: <math>\Leftrightarrow \left(x - \frac18\right)^2 + \left(y - \frac94\right)^2 = \frac{45}{64}.</math>

This equation represents a [[circle]] with center (1/8, 9/4) and radius <math>\tfrac{3}{8}\sqrt{5}</math>. It is the [[circle of Apollonius#Apollonius' definition of a circle|circle of Apollonius]] defined by these values of ''k'', ''A'', and ''B''.