In analytic geometry, positions of points are specified coordinates so that geometrical relationships between the points are equivalent to algebraic relationships between their coordinates. Because of this correspondence between algebra and geometry, it is often possible to prove propositions concerning geometric relationships by means of algebraic calculations. These algebraic techniques have proved very effective.

The invention of analytic geometry is generally credited to the French philosopher and mathematician Rene Descartes (1595-1650). Descartes put down the fundamental principles of analytic geometry in his Discours de la methode (1637). Pierre de Fermat (1601-65) had also worked out the methods of analytic geometry at the same time, but his treatis

The coordinate axes are needed to fix the position of a point in a plane. The point of intersection of these axes is called the origin, denoted by 0. Usually the x-axis is a horizontal line, and the y-axis is the vertical line.

x2+ y2= r2. A conic section is a plane curve that can be represented by a second-degree equation in x and y. The general equation of second degree is

It is always possible to convert from one Cartesian coordinate system into another and also from one Cartesian system to a polar system and vice versa. The path traced by a moving point P(x,y) in the plane is a curve. An equation in two variables x and y that is satisfied by those points on the curve and by no other points is called the equation of the curve. Once the coordinate system has been fixed, a curve has a unique equation, and each equation represents a unique curve, bringing out the connection between algebra and geometry that is supplied by analytic geometry. Any first-degree equation of the form ax + by + c = 0 (where a, b, and c are constants) is the equation of a straight line, or a linear equation.

Riddle, D. F., Analytic Geometry, 4th edition, (1987).

Leonhard Euler (1707- 1783), investigated the general second degree equation in two dimensions, cubic and quartic curves, and introduced the parametric representation of curves, whereby (x,y) are expressed in terms of third variable. For example: a parametric representation of the parabola would be x = at2, y = 2at.

Leithold, Louis, Before Calculus, (1985).

If the x-coordinate of P is x and the y-coordinate of P is y, then the ordered pair (x,y) represents the Cartesian coordinates of P with respect to the fixed coordinate axes. Every ordered pair (x,

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