Coordinate Geometry.

Coordinate Geometry.

1. A number line can be used to represent a number or solution of an equation that only has one variable. It is sufficient to describe the solution of one-valued equations because they all are singledimensional. But as the number of variables in an equation increases, it is not enough.

2. Cartesian Plane: A Cartesian plane is defined by two perpendicular number lines, X and Y. It extends to infinity in both directions. It has a centre usually denoted by O.

   • The horizontal line is called X-axis while the vertical line is called Y-axis.
   • We have seen that two perpendicular number lines X and Y are required for the Cartesian System.
   • The two lines X and Y taken together are called the axes of the system. Note that “axes” is a plural word. If we have to refer to just one of the two number lines X or Y, we will have to use the singular axis.
   • The intersection point of the axes is the zero of the Cartesian System. This point will generally be denoted by O (also known as origin).
   • The two real numbers x and y taken together will describe the position of P uniquely or in an ordered pair. We can write this as follows: P = (x, y). That is, the value along x-axis will be written first, followed by the value along y-axis. And (x, y) ≠ (y, x). For different positions of P, this pair of real numbers will be different.
   • The pair of real numbers x and y which uniquely determine the position of a point P in the plane are referred to collectively as the coordinates of P.

3. The system which we have described to label points in a plane is also known as the Cartesian System. It was first described by a French mathematician named Rene Descartes in the 17th century. The word Cartesian is derived from Descartes.

4. The two axes (X and Y) divide the cartesian plane into four regions, or quadrants

   • In each quadrant, the signs of the coordinates will be different: • First quadrant: x > 0, y > 0 • Second quadrant: x < 0, y > 0 • Third quadrant: x < 0, y < 0 • Fourth quadrant: x > 0, y < 0 Every point on the x-axis is at a distance o unit from the X-axis. So, its ordinate is 0; and the coordinates of such points will be in the form of (x, 0). Every point on the y-axis is at a distance of unit from the Y-axis. So, its abscissa is 0; and the coordinates of such points will be in the form of (0, y). Note: Any point lying on X-axis or Y-axis does not lie in any quadrant.

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