Questions tagged [locus]

A locus is a set of points which satisfy certain geometric conditions.

Many geometric shapes are most naturally and easily described as loci.

For example, a circle is the set of points in a plane which are a fixed distance $~r~$ from a given point $~P~$, the center of the circle.

Problems involving describing a certain locus can often be solved by explicitly finding equations for the coordinates of the points in the locus. Here is a step-by-step procedure for finding plane loci:

Step $1$: If possible, choose a coordinate system that will make computations and equations as simple as possible.

Step $2$: Write the given conditions in a mathematical form involving the coordinates $~x~$ and $~y~$.

Step $3$: Simplify the resulting equations.

Step $4$: Identify the shape cut out by the equations.

Note: Step $~1~$ is often the most important part of the process since an appropriate choice of coordinates can simplify the work in Step $~2~\text{to}~4~$ immensely.

Locus Theorems :

Locus Theorem $1$: The locus of points at a fixed distance, $~d~$, from point $~P~$ is a circle with the given point $~P~$ as its center and $~d~$ as its radius.

Locus Theorem $2$: The locus of points at a fixed distance, $~d~$, from a line, $~l~$, is a pair of parallel lines $~d~$ distance from $~l~$ and on either side of $~l~$.

Locus Theorem $3$: The locus of points equidistant from two points, $~P~$ and $~Q~$, is the perpendicular bisector of the line segment determined by the two points.

Locus Theorem $4$: The locus of points equidistant from two parallel lines, $~l_1~$ and $~l_2~$, is a line parallel to both $~l_1~$ and $~l_2~$ and midway between them.

Locus Theorem $5$: The locus of points equidistant from two intersecting lines, $~l_1~$ and $~l_2~$, is a pair of bisectors that bisect the angles formed by $~l_1~$ and $~l_2~$.

Reference:

https://en.wikipedia.org/wiki/Locus_(mathematics)