EQUATION AND LOCUS

Two fundamental problems of analytic geometry. In this chapter we shall make a preliminary study of the following two fundamental problems of analytic geometry:

I. Given an equation, to determine its geometric interpretation or representation.

II. Given a geometric figure or condition, to determine its equation or analytic representation.

The students will note that these problems are essentially converses of each other. Strictly speaking, however, both problems are so closely related that together they constitute the fundamental problem of all analytic geometry. For example, we shall see later that, after obtaining the equation for a given geometric condition, it is often possible by a study of this equation to determine further geometric characteristics and properties for the given condition. Our purpose in initially considering two separate problems is not one of necessity, but rather one of convenience; we are thus enabled to focus our attention on fewer ideas at a time.

First Fundamental Problem. The Locus of an Equation.

Assume that we are given an equation in the two variables x and y, which we may write briefly in the form

f (x, y) = 0 (1)

In general there are infinitely many pairs of values of x and y which satisfy this equation. Each such pair of real values will be taken as the coordinates (x, y) of a point in the plane.

This convention is the basis of

Definition 1. The totality of points, and only those points, whose coordinates satisfy an equation (1), is called the locus or graph of the equation.

Another convenient expression is given by

Definition 2. Any point whose coordinates satisfy an equation (1) is said to lie on the locus of the equation.

It cannot be emphasized too strongly that only those points whose coordinates satisfy an equation lie on its locus. That is, if the coordinates of a point satisfy an equation, that point lies on the locus of the equation; and conversely, if a point lies on the locus of an equation, its coordinates satisfy the equation. Since the coordinates of the point of a locus are restricted by its equation, such points will in general be located in positions which, taken together, form a definite path called a curve as well as a graph or locus.

Second fundamental problem. We will now consider the second fundamental problem of analytic geometry.

A geometric figure, such as a curve, is generally given by its definition. By the definition of an object is meant a description of that object of such a nature that it is possible to identify it definitely among all other objects of its class. The implication of this statement should be carefully noted: it expresses a necessary and sufficient condition for the existence of the object defined. Thus, let us consider that we are defining a plane curve of type C by means of a unique property P which C possesses. Then, in the entire class of all plane curves, a curve is of type C if and only if it possesses property P.

As a specific example, let us consider that familiar plane curve, the circle. We define a circle as a plane curve possessing the unique property P that all its points are equally distant from a fixed point in its plane. This means that every circle has property P; and conversely, every plane curve having property P is a circle.

For a curve, a geometric condition is a law which the curve must obey. This means that every point on the curve must satisfy the particular law for the curve. Accordingly a curve is often defined as the locus or path traced by a point moving in accordance with a specified law. Thus, a circle may be defined as the locus moving in a plane so that it is always at a constant distance from a fixed point in that plane. A locus need not necessarily satisfy a single condition; it may satisfy two or more conditions. Thus, we may have a curve which is the locus of a point moving so that it passes through a given point, and it is always at a constant distance from a given line. We may then summarize the preceding remarks in the following.

Definition.A curve is the locus of all those points, and only those points, which satisfy one or more given geometric conditions.

The student should note that this definition implies that the given condition or conditions are both necessary and sufficient for the existence of the curve.

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