Hello students, welcome to today's mathematics lesson. I am so happy to be here with you to explore one of the most fascinating and practical branches of mathematics called Coordinate Geometry. Now, before we begin, let me ask you something. Have you ever wondered how we describe where something is located? Think about it. When you tell your friend where your house is, what do you say? You might say something like, "My house is on the second street, and it is the fifth house on that street." Or when you describe where you sit in the classroom, you might say, "I sit in the fifth column and the third row." You see, students, in both these cases, you are using two pieces of information to describe a position. This is exactly what coordinate geometry is all about.
Let me give you another example. Imagine you have a blank sheet of paper, and I put a small dot somewhere on it. Now, if I ask you to tell me where that dot is, how would you describe it? You might say, "It is near the top left corner" or "It is in the upper half of the paper." But you know, these descriptions are not precise. They don't tell us the exact location. However, if you say, "The dot is 5 centimeters from the left edge and 9 centimeters from the bottom," then we know exactly where the dot is! This is because we have used two independent pieces of information: the distance from the left edge and the distance from the bottom edge. Students, this simple idea of using two distances to locate a point is the foundation of coordinate geometry.
Now, let me tell you about a very interesting classroom activity that will help you understand this better. It is called the Seating Plan activity. Imagine all the desks in your classroom are pushed together in rows and columns. Each desk can be represented by a square, and in each square, we write the name of the student sitting there. Now, if I ask you to describe your position in the classroom, you would say something like, "I sit in the fifth column and the third row." This gives us two pieces of information: the column number and the row number. We write this as (5, 3), where the first number is the column and the second is the row. Notice that (5, 3) is not the same as (3, 5)! The order matters very much. If Sonia sits in the fourth column and first row, we write her position as (4, 1).
So students, what have we learned so far? We have learned that to describe the position of any point on a flat surface, we need two independent pieces of information. This could be two distances from two perpendicular lines, or a column number and a row number. This idea is so powerful and important that it has given rise to an entire branch of mathematics called Coordinate Geometry. And today, we are going to learn the basics of this wonderful topic.
Now, let me tell you about the fascinating history of this subject. Coordinate geometry was developed by a great French mathematician and philosopher named René Déscartes, who lived from 1596 to 1650. The story goes that Descartes liked to lie in bed and think. One day, while resting in bed, he solved the problem of describing the position of a point in a plane. He built upon the older idea of latitude and longitude, which sailors used to navigate the seas. In honor of Descartes, the system we use today is called the Cartesian system. Isn't that wonderful, students? A mathematical idea that was conceived while lying in bed!
Now, let us move on to understanding the Cartesian system in detail. You have already studied the number line in your earlier classes. On a number line, we have a fixed point called the origin, from which we measure distances. Points to the right of the origin represent positive numbers, and points to the left represent negative numbers. For example, the point at a distance of 3 units to the right of the origin represents the number 3, and the point at a distance of 2 units to the left represents the number -2.
Descartes had a brilliant idea. He took two such number lines and placed them perpendicular to each other. One line is horizontal, and the other is vertical. The horizontal line is called the x-axis, and the vertical line is called the y-axis. The point where these two axes intersect is called the origin, and it is denoted by the letter O. The positive direction of the x-axis is to the right, and the positive direction of the y-axis is upward. Similarly, the negative direction of the x-axis is to the left, and the negative direction of the y-axis is downward.
Now, students, when we draw these two perpendicular axes on a plane, they divide the plane into four parts. Each part is called a quadrant, which means one-fourth. These quadrants are numbered I, II, III, and IV, in an anticlockwise direction starting from the positive x-axis. So, the first quadrant is between the positive x-axis and the positive y-axis, the second quadrant is between the negative x-axis and the positive y-axis, the third quadrant is between the negative x-axis and the negative y-axis, and the fourth quadrant is between the positive x-axis and the negative y-axis. Together, the axes and these quadrants form what we call the Cartesian plane, or the coordinate plane, or the xy-plane.
Now, let us learn how to describe the position of any point in this plane. Suppose we have a point P in the first quadrant. To describe its position, we draw a perpendicular from P to the x-axis, and let us call the point where this perpendicular meets the x-axis as M. We also draw a perpendicular from P to the y-axis, and let the meeting point be N. Now, the distance of P from the y-axis, which is the length of the perpendicular PN or OM, is called the x-coordinate or abscissa of point P. And the distance of P from the x-axis, which is the length of the perpendicular PM or ON, is called the y-coordinate or ordinate of point P. For point P in our example, let us say that the distance from the y-axis is 4 units and the distance from the x-axis is 3 units. Then the x-coordinate or abscissa is 4, and the y-coordinate or ordinate is 3. We write the coordinates of point P as (4, 3). Remember, students, the x-coordinate always comes first, and the y-coordinate comes second. This is very important.
Now, let us consider another point Q in the third quadrant. If Q is 6 units to the left of the y-axis and 2 units below the x-axis, then its x-coordinate is -6 and its y-coordinate is -2. So we write the coordinates of Q as (-6, -2). Notice how the signs tell us which direction the point is from the axes.
Students, let me summarize what we have learned so far. The x-coordinate or abscissa of a point is its perpendicular distance from the y-axis, measured along the x-axis. It is positive when measured in the positive direction of the x-axis and negative when measured in the negative direction. Similarly, the y-coordinate or ordinate of a point is its perpendicular distance from the x-axis, measured along the y-axis. It is positive in the positive direction of the y-axis and negative in the negative direction. And when we write coordinates, we always write the x-coordinate first, then the y-coordinate, in brackets.
Now, let us look at some special cases. What about points that lie on the axes themselves? Consider a point on the x-axis. Since it lies on the x-axis, its distance from the x-axis is zero. So its y-coordinate is 0. For example, a point that is 4 units to the right of the origin on the x-axis has coordinates (4, 0). Similarly, a point on the y-axis has coordinates (0, y), where y is its distance from the x-axis. And what about the origin itself? The origin is at zero distance from both axes, so its coordinates are (0, 0). These are very important points to remember.
Now, let us discuss the relationship between the signs of coordinates and the quadrant in which a point lies. This is a very useful pattern that will help you quickly identify where a point is located. In the first quadrant, both x and y are positive, so coordinates are of the form (+, +). In the second quadrant, x is negative and y is positive, so coordinates are of the form (-, +). In the third quadrant, both x and y are negative, so coordinates are of the form (-, -). And in the fourth quadrant, x is positive and y is negative, so coordinates are of the form (+, -). This pattern is like a map that tells you which quadrant you are in just by looking at the signs of the coordinates.
Let me now work through some examples with you so that you can see how all these concepts are applied.
Consider Example 1 from your textbook. We are given a figure with several points labeled B, M, L, and S. We need to find their coordinates.
For point B, we see that its distance from the y-axis is 4 units, so its x-coordinate or abscissa is 4. Its distance from the x-axis is 3 units, so its y-coordinate or ordinate is 3. Therefore, the coordinates of point B are (4, 3).
For point M, its distance from the y-axis is 3 units to the left, so its x-coordinate is -3. Its distance from the x-axis is 4 units upward, so its y-coordinate is 4. Therefore, the coordinates of point M are (-3, 4).
For point L, its distance from the y-axis is 5 units to the left, so its x-coordinate is -5. Its distance from the x-axis is 4 units downward, so its y-coordinate is -4. Therefore, the coordinates of point L are (-5, -4).
For point S, its distance from the y-axis is 3 units to the right, so its x-coordinate is 3. Its distance from the x-axis is 4 units downward, so its y-coordinate is -4. Therefore, the coordinates of point S are (3, -4).
Notice how each point falls into a different quadrant: B is in the first quadrant, M is in the second quadrant, L is in the third quadrant, and S is in the fourth quadrant. This matches exactly with the sign pattern we discussed earlier.
Now, let us look at Example 2. We are asked to write the coordinates of points that lie on the axes. Let us consider point A, which is on the x-axis at a distance of 4 units from the y-axis. Since it is on the x-axis, its distance from the x-axis is zero. So its coordinates are (4, 0). Similarly, point B is on the y-axis at a distance of 3 units from the x-axis, so its coordinates are (0, 3). Point C is on the x-axis at a distance of 5 units to the left of the y-axis, so its coordinates are (-5, 0). Point D is on the y-axis at a distance of 4 units below the x-axis, so its coordinates are (0, -4). And point E is on the x-axis at a distance of 2/3 units from the y-axis, so its coordinates are (2/3, 0).
From these examples, we can see an important pattern. Any point on the x-axis has coordinates of the form (x, 0), where x is the distance from the y-axis. Any point on the y-axis has coordinates of the form (0, y), where y is the distance from the x-axis. And the origin, being at the intersection of both axes, has coordinates (0, 0).
Now, students, I want you to remember one more very important thing. The order of coordinates matters a great deal. The point (3, 4) is completely different from the point (4, 3). (3, 4) means 3 units from the y-axis and 4 units from the x-axis, while (4, 3) means 4 units from the y-axis and 3 units from the x-axis. They are two different points in the plane. Only when x equals y, the coordinates (x, y) and (y, x) represent the same point. For example, (3, 3) and (3, 3) are the same point.
Let me now recap what we have learned in this chapter so that it stays clear in your mind.
First, we learned that to locate the position of a point in a plane, we need two perpendicular lines. One is horizontal, called the x-axis, and the other is vertical, called the y-axis. Together, they form the Cartesian plane or coordinate plane.
Second, the point where the two axes intersect is called the origin, and it is denoted by O. Its coordinates are (0, 0).
Third, the axes divide the plane into four parts called quadrants, numbered I, II, III, and IV in an anticlockwise direction.
Fourth, the distance of a point from the y-axis is called its x-coordinate or abscissa, and the distance from the x-axis is called its y-coordinate or ordinate. Together, they form the coordinates of the point, written as (x, y).
Fifth, points on the x-axis have coordinates of the form (x, 0), and points on the y-axis have coordinates of the form (0, y).
Sixth, the signs of the coordinates tell us which quadrant the point lies in: first quadrant (+, +), second quadrant (-, +), third quadrant (-, -), and fourth quadrant (+, -).
And finally, the coordinates (x, y) and (y, x) are different unless x equals y.
Students, coordinate geometry is a powerful tool that you will use in many different areas of mathematics and in real life. It allows us to describe positions precisely, to draw graphs, to solve problems, and much more. What we have learned today is just the beginning. You will explore this topic in much greater detail in your higher classes.
Now, before we end this lesson, let me summarize everything we have covered in this chapter.
In this chapter on Coordinate Geometry, we have studied the following important points:
We learned that to locate the position of an object or a point in a plane, we require two perpendicular lines. One is horizontal, and the other is vertical.
The plane is called the Cartesian plane, or coordinate plane, or xy-plane, and the lines are called the coordinate axes.
The horizontal line is called the x-axis, and the vertical line is called the y-axis.
The coordinate axes divide the plane into four parts called quadrants.
The point of intersection of the axes is called the origin.
The distance of a point from the y-axis is called its x-coordinate, or abscissa, and the distance of the point from the x-axis is called its y-coordinate, or ordinate.
If the abscissa of a point is x and the ordinate is y, then (x, y) are called the coordinates of the point.
The coordinates of a point on the x-axis are of the form (x, 0), and that of a point on the y-axis are (0, y).
The coordinates of the origin are (0, 0).
The coordinates of a point are of the form (positive, positive) in the first quadrant, (negative, positive) in the second quadrant, (negative, negative) in the third quadrant, and (positive, negative) in the fourth quadrant.
If x is not equal to y, then (x, y) is not equal to (y, x), and (x, y) equals (y, x) only if x equals y.
Students, you have done wonderfully well today. I am so proud of you for learning all these concepts carefully. Remember, practice is the key to mastering coordinate geometry. Try to identify coordinates of points in your daily life, and you will see how useful this knowledge is. Thank you for being such an attentive audience. Keep practicing, and until next time, goodbye!