軌跡（数学）

Find the locus of a point P that has a given ratio of distances k = d₁/d₂ 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

${\displaystyle \Leftrightarrow |PA|=3|PB|}$

${\displaystyle \Leftrightarrow |PA|^{2}=9|PB|^{2}}$

${\displaystyle \Leftrightarrow (x+1)^{2}+(y-0)^{2}=9(x-0)^{2}+9(y-2)^{2}}$

${\displaystyle \Leftrightarrow 8(x^{2}+y^{2})-2x-36y+35=0}$

${\displaystyle \Leftrightarrow \left(x-{\frac {1}{8}}\right)^{2}+\left(y-{\frac {9}{4}}\right)^{2}={\frac {45}{64}}.}$

This equation represents a circle with center (1/8, 9/4) and radius ${\displaystyle {\tfrac {3}{8}}{\sqrt {5}}}$. It is the circle of Apollonius defined by these values of k, A, and B.

A triangle ABC has a fixed side [AB] with length c. Determine the locus of the third vertexC such that the medians from A and C are orthogonal.

Choose an orthonormalcoordinate system such that A(−c/2, 0), B(c/2, 0). C(x, y) is the variable third vertex. The center of [BC] is M((2x + c)/4, y/2). The median from C has a slope y/x. The median AM has slope 2y/(2x + 3c).

C(x, y) is a point of the locus

${\displaystyle \Leftrightarrow }$ the medians from A and C are orthogonal

${\displaystyle \Leftrightarrow {\frac {y}{x}}\cdot {\frac {2y}{2x+3c}}=-1}$

${\displaystyle \Leftrightarrow 2y^{2}+2x^{2}+3cx=0}$

${\displaystyle \Leftrightarrow x^{2}+y^{2}+(3c/2)x=0}$

${\displaystyle \Leftrightarrow (x+3c/4)^{2}+y^{2}=9c^{2}/16.}$

The locus of the vertex C is a circle with center (−3c/4, 0) and radius 3c/4.

A locus can also be defined by two associated curves depending on one common parameter. If the parameter varies, the intersection points of the associated curves describe the locus.

In the figure, the points K and L are fixed points on a given line m. The line k is a variable line through K. The line l through L is perpendicular to k. The angle ${\displaystyle \alpha }$ between k and m is the parameter. k and l are associated lines depending on the common parameter. The variable intersection point S of k and l describes a circle. This circle is the locus of the intersection point of the two associated lines.

A locus of points need not be one-dimensional (as a circle, line, etc.). For example,^[1] the locus of the inequality 2x + 3y – 6 < 0 is the portion of the plane that is below the line of equation 2x + 3y – 6 = 0.