Coordinate Geometry

Coordinate geometry is an essential branch of mathematics that acts as a bridge between algebra and geometry. It was developed by the French philosopher and mathematician René Descartes. He realized that a point in a plane could be described exactly by a pair of numbers, using two perpendicular lines as a reference system.

In this chapter, we will learn about the Cartesian plane, coordinates of a point, and how to plot points on a graph.

1. The Cartesian System

The system used for describing the position of a point in a plane is called the Cartesian system, named in honor of René Descartes.

To locate the position of an object or a point in a plane, we require two mutually perpendicular lines. One of them is horizontal, and the other is vertical.

• The horizontal line is called the x-axis (denoted by $XX'$).
• The vertical line is called the y-axis (denoted by $YY'$).
• The point where the x-axis and the y-axis intersect is called the origin and is denoted by $O$.

Positive and Negative Directions

• The directions $OX$ and $OY$ are called the positive directions of the x-axis and y-axis, respectively.
• The directions $OX'$ and $OY'$ are called the negative directions of the x-axis and y-axis, respectively.

2. Quadrants

The two axes divide the plane into four regions. These four regions are called quadrants (one-fourth part). We number the quadrants I, II, III, and IV anticlockwise starting from the $OX$ and $OY$ (positive directions).

• Quadrant I: Bounded by positive x-axis and positive y-axis.
• Quadrant II: Bounded by negative x-axis and positive y-axis.
• Quadrant III: Bounded by negative x-axis and negative y-axis.
• Quadrant IV: Bounded by positive x-axis and negative y-axis.

The entire plane, consisting of the axes and the quadrants, is known as the Cartesian plane, coordinate plane, or xy-plane.

3. Coordinates of a Point

To find the coordinates of a point $P$ in the Cartesian plane, we draw perpendiculars from $P$ to the x-axis and the y-axis.

1. Abscissa (x-coordinate): The x-coordinate of a point is its perpendicular distance from the y-axis. This is called the abscissa.
2. Ordinate (y-coordinate): The y-coordinate of a point is its perpendicular distance from the x-axis. This is called the ordinate.

When writing the coordinates of a point in a plane, we enclose them in parentheses and separate them by a comma. The x-coordinate always comes first, followed by the y-coordinate. Format: $(x, y)$

Examples:

• If the perpendicular distance of a point from the y-axis is $4$ units in the positive direction, its x-coordinate is $4$.
• If its perpendicular distance from the x-axis is $3$ units in the negative direction, its y-coordinate is $-3$.
• The coordinates of this point are $(4, -3)$.

Special Coordinates:

• The coordinates of the origin $O$ are $(0, 0)$.
• The coordinates of a point on the x-axis are of the form $(x, 0)$. Since its perpendicular distance from the x-axis is zero, the ordinate is always zero.
• The coordinates of a point on the y-axis are of the form $(0, y)$. Since its perpendicular distance from the y-axis is zero, the abscissa is always zero.

4. Plotting a Point in the Plane

If we are given the coordinates of a point, we can locate it in the Cartesian plane.

Example: Plot the point $P(3, 5)$. 1. Start at the origin $O(0,0)$. 2. Move $3$ units along the positive x-axis (to the right). 3. From there, move $5$ units parallel to the positive y-axis (upwards). 4. Mark the final position as point $P(3, 5)$.

Example: Plot the point $Q(-2, 4)$. 1. Start at the origin $O$. 2. Move $2$ units along the negative x-axis (to the left). 3. From there, move $4$ units parallel to the positive y-axis (upwards). 4. Mark the final position as point $Q(-2, 4)$.

Signs of Coordinates in Quadrants:

By looking at the signs of the x and y coordinates, we can immediately identify which quadrant a point lies in:

• Quadrant I: $(+, +)$ Both abscissa and ordinate are positive (e.g., $3, 2$).
• Quadrant II: $(-, +)$ Abscissa is negative, ordinate is positive (e.g., $-4, 5$).
• Quadrant III: $(-, -)$ Both abscissa and ordinate are negative (e.g., $-1, -6$).
• Quadrant IV: $(+, -)$ Abscissa is positive, ordinate is negative (e.g., $7, -2$).

5. Applications of Coordinate Geometry

Coordinate geometry is not just for plotting points; it has extensive real-world and mathematical applications. * Map Reading: Maps use a grid system (latitude and longitude) to define locations on Earth. * Computer Graphics: Every pixel on a screen has an $(x, y)$ coordinate. Software uses these coordinates to draw lines, shapes, and render images. * Physics and Engineering: Determining the paths of projectiles, orbits of planets, and positions in 3D space rely heavily on coordinate systems. * Algebraic Geometry: It allows us to graph algebraic equations and analyze geometric figures (like circles, parabolas, and lines) algebraically.

Summary

The Cartesian plane is a two-dimensional number line grid. Understanding abscissa, ordinate, and quadrants allows us to precisely locate any point in space. This foundational knowledge is crucial for graphing linear equations and delving deeper into analytical geometry in higher classes.