Analytic Geometry

Analytic Geometry, branch of geometry in which straight lines, curves, and geometric figures are represented by numerical and algebraic expressions using a set of axes and coordinates. Any point in a plane may be located with respect to a pair of perpendicular axes by specifying the distance of the point from each of these axes. In figure 1, point a is 1 unit from the vertical y-axis and 4 units from the horizontal x-axis. The coordinates of point a are 1 and 4, and the point is located by the statements x = 1, y = 4. Positive x values are located to the right of the y-axis and negative values to the left; positive y values are above the x-axis and negative y values below. Thus, point b in figure 1 has the coordinates x = 5, y = 0. Points in three-dimensional space can be similarly located with respect to three axes, of which the third, usually called the z-axis, is perpendicular to the other two at their point of intersection, which is called the origin.

In general, a straight line can always be represented by a linear equation in two variables, x and y, in the form ax + by + c = 0. In the same way, equations can be derived for the circle, ellipse, and other conic sections and regular curves. The problems treated in analytic geometry are of two classic kinds. The first is: given a geometric description of a set of points, determine the algebraic equation that is satisfied by these points. In the above example, the collection of points that lies on the straight line passing through the points a and b satisfies the linear equation x + y = 5. In general, ax + by = c. The second kind of problem is: given an algebraic statement, describe the locus of the points that satisfy the statement in geometric terms. For example, a circle of radius 3 and with its centre at the origin is the locus of points that satisfy the equation x² + y² = 9.