# 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 i.e., 1)every point satisfies a given condition and 2)every point satisfiying it is in that particular locus

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

Examples from plane geometry:

• The set of points equidistant from two lines which cross is the angle bisector.
• All conic sections are loci:
• Parabola: the set of points equidistant from a single point (the focus) and a line (the directrix).
• Circle: the set of points for which the distance from a single point is constant (the radius). The set of points for each of which the ratio of the distances to two given foci is a positive constant (that is not 1) is referred to as a Circle of Apollonius.
• 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.

## Proof of a Locus

In order to prove the correctness of a locus, one generally divides the proof into two stages:

• Proof that all the points that satisfy the conditions are on the locus.
• Proof that all the points on the locus satisfy the conditions.

## Example

C

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

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.