Coordinate Geometry - Cartesian System

The Cartesian Plane is a two-dimensional surface formed by the intersection of two perpendicular number lines. The horizontal line is called the $x$-axis ($X'OX$) and the vertical line is called the $y$-axis ($Y'OY$). Their point of intersection is called the Origin, denoted by $O(0, 0)$. Visually, this creates a grid where the $x$-axis acts like a horizontal ruler and the $y$-axis acts like a vertical ruler.

The two axes divide the plane into four distinct regions called Quadrants, numbered I, II, III, and IV in a counter-clockwise direction starting from the top-right. Imagine the plane split into four 'rooms': top-right (I), top-left (II), bottom-left (III), and bottom-right (IV).

Any point in the plane is identified by an ordered pair of numbers $(x, y)$, known as coordinates. The first number, $x$, is the Abscissa (the perpendicular distance from the $y$-axis). The second number, $y$, is the Ordinate (the perpendicular distance from the $x$-axis).

Sign Convention of Quadrants: In Quadrant I, both coordinates are positive $(+, +)$. In Quadrant II, the $x$-coordinate is negative and the $y$-coordinate is positive $(-, +)$. In Quadrant III, both coordinates are negative $(-, -)$. In Quadrant IV, the $x$-coordinate is positive and the $y$-coordinate is negative $(+, -)$. This determines where a point is plotted relative to the origin.

Points on the Axes: If a point lies on the $x$-axis, its $y$-coordinate is always $0$, represented as $(x, 0)$. If a point lies on the $y$-axis, its $x$-coordinate is always $0$, represented as $(0, y)$. These points do not belong to any specific quadrant but serve as the boundaries between them.

Plotting a point $(a, b)$ involves a specific movement sequence: Start at the origin $(0, 0)$. Move $|a|$ units along the $x$-axis (right if $a$ is positive, left if $a$ is negative). From that position, move $|b|$ units parallel to the $y$-axis (up if $b$ is positive, down if $b$ is negative). The final location is the point $(a, b)$.