Locus (mathematics)

In Geometry, a Locus (plural: loci) is a set of points whose location satisfies or is determined by one or more specified conditions.

Each curve in this example is the locus of a set of points that lie on any line defined as the conchoid of a circle centered at point P and the line l. In this example, P is 7cm from l.

Commonly studied loci

All conic sections are all loci:

• Parabola: the set of points equidistant from a single point (the focus) and a line (the directrix).
• Hyperbola: the set of points for each of which the absolute value of the difference between the distances to two given foci is a constant.
• Ellipse: the set of points for each of which the sum of the distances to two given foci is a constant. In particular, the circle is a locus.

Straight lines are loci too: every straight line is the set of all points equidistant to two given points.

Example

Consider the locus of all points P such that its distance from the point (−1,0) is three times its distance from the point (0,2). If P = (x,y), then, saying that P belongs to the locus means that

$\sqrt{(x+1)^2+(y-0)^2}=3\sqrt{(x-0)^2+(y-2)^2}$

On squaring,

$(x+1)^2+(y-0)^2=9(x-0)^2+9(y-2)^2\Leftrightarrow 8(x^2+y^2)-2x-36y+35 =0$

This is equivalent to

$\left(x-\frac18\right)^2+\left(y-\frac94\right)^2=\frac{45}{64}$

This equation represents a circle.

See also

• Conic section
• Set (mathematics)

References

• Robert Clarke James, Glenn James: Mathematics Dictionary. Springer 1992, ISBN 978-0-412-99041-0, p. 255 (restricted online copy at Google Books)
• Alfred North Whitehead: An Introduction to Mathematics. BiblioBazaar LLC 2009 (reprint), ISBN 978-1-103-19784-2, pp. 121 (restricted online copy at Google Books)
• George Wentworth: Junior High School Mathematics: Book III. BiblioBazaar LLC 2009 (reprint), ISBN 978-1-103-15236-0, pp. 265 (restricted online copy at Google Books)