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 from a given point 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 and .
Step 3: Simplify the resulting equations.
Step 4: Identify the shape cut out by the equations.
Step 1 is often the most important part of the process since an appropriate choice of coordinates can simplify the work in steps 2-4 immensely.
Find the locus of points such that the sum of the squares of the distances from to and from to where and are two fixed points in the plane, is a fixed positive constant.
After rotating and translating the plane, we may assume that and Suppose the constant is Then
So the locus is either empty if a point if or a circle if
Describe the locus of the points in a plane which are equidistant from a line and a fixed point not on the line.
After rotation and translation (and possibly reflection), we may assume that the point is with and that the line is the -axis. The distance from to the -axis is and the distance to the point is so the equation becomes
which describes a parabola.
Note that if the point did lie on the line, e.g. the equation reduces to or which gives a line perpendicular to the original line through the point; this makes sense geometrically as well.
Find the locus of all points in a plane such that the sum of the distances and is a fixed constant, where and are two fixed points in the plane.
After translating and rotating, we may assume and and let the constant be If then the locus is clearly empty, and if then the locus is a point, so assume Let and Then and The locus equation is
Since and this is the equation of an ellipse.
Note that if this describes a circle, as expected and coincide