Coordinate Geometry — Study Notes

1. Introduction

Coordinate geometry, also called analytic geometry, is the branch of mathematics in which we use a pair of numbers to describe the exact position of a point in a plane. This beautiful idea was developed by the French mathematician Rene Descartes in the seventeenth century. Before his work, geometry and algebra were studied as completely separate subjects. Descartes showed that a line or a curve could be represented by an equation, and an equation could be visualised as a curve. This union of algebra and geometry revolutionised mathematics.

The idea of coordinates is used everywhere in daily life. When we give a seat number in a cinema hall, we say row number and seat number, which is exactly a coordinate pair. A location on a map is given by latitude and longitude. In a chess game, every square is identified by a letter and a number. In this chapter we will learn how to set up the coordinate plane, how to read the coordinates of a point, how to plot points, and how to identify the quadrants in which points lie.

2. The Cartesian Plane

The Cartesian plane is formed by drawing two mutually perpendicular lines that intersect at a common point. These two lines are called the coordinate axes.

• The horizontal line is called the x-axis.
• The vertical line is called the y-axis.
• The point of intersection of the two axes is called the origin, usually denoted by O.
• The two axes divide the plane into four parts, called quadrants, numbered I, II, III and IV in the anticlockwise direction.

The Axes in Detail

The x-axis is the set of all points whose y-coordinate is zero, and the y-axis is the set of all points whose x-coordinate is zero. The origin is the point (0, 0). The positive direction on the x-axis is towards the right of the origin, and the positive direction on the y-axis is upwards. The negative x-direction is to the left, and the negative y-direction is downwards.

3. Coordinates of a Point

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

• The distance of P from the y-axis (measured along the horizontal) is called the x-coordinate or the abscissa of P.
• The distance of P from the x-axis (measured along the vertical) is called the y-coordinate or the ordinate of P.
• The coordinates are written as an ordered pair (x, y), with the abscissa written first and the ordinate second.

The word "ordered" is very important: (3, 5) and (5, 3) are two completely different points. The abscissa tells us how far to move to the right (if positive) or left (if negative) from the origin, and the ordinate tells us how far to move up (if positive) or down (if negative).

Signs of Coordinates in the Quadrants

• In quadrant I: x is positive and y is positive, written (+, +).
• In quadrant II: x is negative and y is positive, written (-, +).
• In quadrant III: x is negative and y is negative, written (-, -).
• In quadrant IV: x is positive and y is negative, written (+, -).

A point on the x-axis has coordinates (x, 0), and a point on the y-axis has coordinates (0, y). The origin is (0, 0).

4. Plotting Points on the Cartesian Plane

To plot the point (x, y) on the coordinate plane, we follow these steps: 1. Start from the origin O. 2. Move along the x-axis by the amount x: to the right if x is positive, to the left if x is negative. 3. From that position, move parallel to the y-axis by the amount y: upwards if y is positive, downwards if y is negative. 4. Mark the final position with a dot and label it with its coordinates.

For example, to plot (4, 3), move 4 units to the right along the x-axis and then 3 units up. To plot (-2, -5), move 2 units to the left and then 5 units down.

Reading Coordinates from a Graph

To read the coordinates of a point already drawn on a graph, we drop perpendiculars to both axes. The value where the perpendicular meets the x-axis is the abscissa, and the value where the perpendicular meets the y-axis is the ordinate. Care must be taken to read the values on the correct scales of the axes.

5. Equations and Graphs

In later chapters we will learn that an equation like 2x + 3y = 6 represents a straight line, and every point whose coordinates satisfy the equation lies on that line. Coordinate geometry provides the bridge between such equations and their geometric pictures. The x-coordinate of a point on the graph is the input value, and the y-coordinate is the corresponding output value.

The Distance Between Points

A useful result derived from coordinates is the distance formula. The distance between two points A(x1, y1) and B(x2, y2) is given by: distance AB = sqrt ((x2 - x1)^2 + (y2 - y1)^2) This formula is obtained by applying the Pythagoras theorem to the right triangle formed by the two points. For example, the distance between A(1, 2) and B(4, 6) is sqrt ((4 - 1)^2 + (6 - 2)^2) = sqrt (9 + 16) = sqrt 25 = 5 units. The distance of a point P(x, y) from the origin O(0, 0) is sqrt (x^2 + y^2).

Quick Revision Tables

Table 1: Signs of Coordinates in Quadrants

Table 2: Important Points and Their Coordinates

Mind Map

Important Diagrams (SVG)

Diagram 1: The Cartesian Plane with Quadrants

Diagram 2: Distance Between Two Points

Common Mistakes

1. Students often write the coordinates in the wrong order. The abscissa (x-coordinate) must always be written first: (x, y), so (3, 5) is not the same point as (5, 3).
2. A very common error is getting the signs of the quadrants wrong. Remember the sequence: I(+, +), II(-, +), III(-, -), IV(+, -), going anticlockwise.
3. Many students believe that the point (0, 5) lies on the x-axis. In fact, a point with x-coordinate 0 lies on the y-axis.
4. When plotting a negative point like (-3, -2), students sometimes move in the wrong direction on one axis. Both movements are opposite to the positive direction.
5. Students forget to mention that the coordinates are measured as perpendicular distances, and instead measure along the axes at an angle.
6. In the distance formula, students forget to square the differences or forget to take the square root of the final sum, giving the square of the distance.
7. Students confuse the coordinate of the origin, sometimes writing (1, 1). The origin is always (0, 0).

Exam Tips

1. Always draw a rough sketch of the coordinate plane before plotting points; a neat figure with labelled axes earns credit in the exam.
2. To identify the quadrant, first check the sign of x (left or right) and then the sign of y (up or down); do not mix the two.
3. Memorise the rule "x-axis points have y = 0 and y-axis points have x = 0" because it appears frequently in one-mark questions.
4. In the distance formula, always write the two points clearly as (x1, y1) and (x2, y2) before substituting, to avoid sign errors.
5. Check your answer by verifying that a point on the x-axis, like (4, 0), is plotted by moving 4 units right and 0 units up.
6. Practise plotting points from all four quadrants and from the axes, since these are the most common examination questions.
7. When a question says the point lies "on the x-axis", immediately set the y-coordinate to 0 and then solve for x.

Conclusion

In this chapter we learnt how to describe the position of a point in a plane using an ordered pair of numbers, called its coordinates. We studied the Cartesian plane formed by the x-axis and the y-axis meeting at the origin, and the four quadrants with their characteristic signs of coordinates. We learnt how to plot a given point and how to read the coordinates of a point from a graph. We also derived the distance formula using the Pythagoras theorem, which lets us calculate the distance between any two points in the plane. Coordinate geometry is the meeting point of algebra and geometry, and it will be used extensively in the next chapter on linear equations in two variables and in many later classes, so mastering the basics here is essential.