The equation of a locus. We shall now study the problem of determining the equation of a locus as the analytic interpretation of the geometric condition or conditions defining the

We shall now study the problem of determining the equation of a locus as the analytic interpretation of the geometric condition or conditions defining the locus.

Definition. The equation of a plane locus is an equation

f (x, y) = 0 (1)

all whose corresponding real solutions for x and y are the coordinates of those points, and only those points, satisfying the given geometric condition or conditions which define the locus.

Note that this definition expresses a necessary and sufficient condition that (1) be the equation of a locus. Accordingly, the procedure in obtaining the equation of a locus is essentially as follows:

Let the point P with coordinates (x, y) be any point satisfying the given condition or conditions, and hence a point on the locus.

Express the given geometric condition or conditions analytically by means of an equation or equations in the variable coordinates x and y.

Simplify, if necessary, the equation obtained in Step 2 so that it assumes the form (1).

Conversely, let (x₁, y₁) be the coordinates of any point satisfying (1) so that the equation

f (x₁, y₁) = 0 (2)

holds. If (2) leads to the analytic expression for the given geometric condition or conditions, as applied to the point (x₁, y₁), then (1) is the required equation of the locus.

In practice, Step 4 may usually be omitted, since retracing the work from Step 3 back to Step 2 is generally immediate. Note in Step 1 that, by considering P as any point on the locus, we are thereby considering every point on the locus.

II. Topics for discussion:

1. The equation of a plane locus.

The procedure in obtaining the equation of a locus.

T E X T V

GRAMMAR: THE INFINITIVE CONSTRUCTIONS (COMPLEX SUBJECT, COMPLEX OBJECT AND FOR-PHRASES)