coordinate geometry

System of geometry in which points, lines, shapes, and surfaces are represented by algebraic expressions. In plane (two-dimensional) coordinate geometry, the plane is usually defined by two axes at right angles to each other, the horizontal x-axis and the vertical y-axis, meeting at O, the origin. A point on the plane can be represented by a pair of Cartesian coordinates, which define its position in terms of its distance along the x-axis and along the y-axis from O. These distances are, respectively, the x and y coordinates of the point.

Lines are represented as equations; for example, y = 2x + 1 gives a straight line, and y = 3x² + 2x gives a parabola (a curve). The graphs of varying equations can be drawn by plotting the coordinates of points that satisfy their equations, and joining up the points. One of the advantages of coordinate geometry is that geometrical solutions can be obtained without drawing but by manipulating algebraic expressions. For example, the coordinates of the point of intersection of two straight lines can be determined by finding the unique values of x and y that satisfy both of the equations for the lines, that is, by solving them as a pair of simultaneous equations. The curves studied in simple coordinate geometry are the conic sections (circle, ellipse, parabola, and hyperbola), each of which has a characteristic equation.