Points, Axes, and the Cartesian Plane
Plot Any Point, Name Any Quadrant — The Cartesian World Made Simple
Coordinate Geometry in Class IX is the foundation for the more advanced work in Class X. CBSE Boards typically assign 2–3 marks from this chapter through identification of quadrants, coordinates of points, and plotting problems. This chapter is completely scoring — every student should target full marks. NTSE uses coordinate plane questions in its mathematical reasoning section.
This is a completely scoring chapter with minimal content — spend 1 day and aim for perfect marks. Memorise the sign convention for each quadrant: All Students Take Calculus (All +, Sin/y +, Tan/x+y −, Cos/x +). For CBSE, graph-based questions require neat, labelled diagrams with a ruler. Time investment: 1 day.
The French mathematician and philosopher René Descartes introduced the idea of representing geometrical figures using numbers and equations. His revolutionary work created the Cartesian coordinate system, which became the foundation of modern analytical geometry.
Because of his contribution, the coordinate plane is often called the Cartesian Plane.
A Cartesian Plane is formed by two mutually perpendicular number lines:
These axes intersect at a fixed point called the Origin.
\[\small O(0,0) \]
The X-axis and Y-axis divide the coordinate plane into four regions called quadrants.
| Quadrant | Sign of x | Sign of y | Example |
|---|---|---|---|
| I Quadrant | + | + | \(\small (3,2)\) |
| II Quadrant | - | + | \(\small (-4,5)\) |
| III Quadrant | - | - | \(\small (-2,-3)\) |
| IV Quadrant | + | - | \(\small (5,-2)\) |
Plot the point:
\[\small A(3,2) \]
Positive x-coordinate means move right. Positive y-coordinate means move upward.
Therefore, the point \(\small A(3,2)\) lies in the first quadrant.
A drone is flying over a city. Its position is represented by the point \(\small (4,5)\) on a coordinate plane where the x-coordinate shows east-west movement and the y-coordinate shows north-south movement.
In elementary mathematics, we study the number line where numbers are represented on a straight line. Extending this concept to two dimensions allows us to locate positions of objects and points on a flat surface called a plane. The mathematical system developed for locating points on a plane is known as the Cartesian Coordinate System.
This system uses two mutually perpendicular number lines called coordinate axes. Every point on the plane is represented using an ordered pair of numbers known as coordinates. The Cartesian system forms the foundation of graph plotting, analytical geometry, engineering drawings, computer graphics, navigation systems, and scientific data representation.
The Cartesian Coordinate System was introduced by the French mathematician and philosopher René Descartes. His revolutionary idea connected algebra with geometry by allowing geometrical figures to be represented using equations and coordinates.
Because of his contribution, the coordinate plane is often called the Cartesian Plane.
The Cartesian system consists of two perpendicular number lines:
These two axes intersect at a fixed point called the Origin.
\[\small O(0,0) \]
| Axis | Positive Direction | Negative Direction |
|---|---|---|
| X-axis | \(\small OX\) | \(\small OX'\) |
| Y-axis | \(\small OY\) | \(\small OY'\) |
The plane formed by the intersection of the x-axis and y-axis is called the Cartesian Plane or XY-plane.
Every point on this plane is uniquely represented by an ordered pair:
\[\small (x,y) \]
where:
The x-axis and y-axis divide the plane into four regions called Quadrants.
These quadrants are numbered in an anticlockwise direction starting from the upper-right region.
Determine the signs of x-coordinate and y-coordinate.
A robot moves on a coordinate plane. It starts from the origin and moves 5 units right and 4 units upward.
Every point on the Cartesian plane is represented by an ordered pair of numbers written in the form:
\[\small (x,y) \]
These numbers are called the coordinates of the point. Coordinates help us determine the exact location of a point on the Cartesian plane.
The coordinates of a point are written as an Ordered Pair because the order of the numbers is extremely important.
\[\small (x,y)\neq(y,x) \]
Consider the point:
\[\small (3,5) \]
But the point:
\[\small (5,3) \]
represents an entirely different location on the coordinate plane.
The two coordinates of a point have special mathematical names.
| Coordinate | Mathematical Term | Meaning |
|---|---|---|
| \(\small x\)-coordinate | Abscissa | Horizontal distance from the y-axis |
| \(\small y\)-coordinate | Ordinate | Vertical distance from the x-axis |
The abscissa is the horizontal distance of a point from the y-axis.
It tells us how far the point is located towards the right or left side.
The ordinate is the vertical distance of a point from the x-axis.
It tells us how far the point is located upward or downward.
The point \(\small (x,y)\) is obtained by:
In the ordered pair \(\small (x,y)\):
Abscissa: \[\small 7\]
Ordinate: \[\small -3\]
Therefore, the point lies in the:
Second Quadrant
A treasure is hidden at the point: \[\small -6,5)\] on a coordinate map.
Point \(\small B\) lies:
Therefore:
Abscissa \(\small =4\)
Ordinate \(\small =3\)
Hence, the coordinates of \(\small B\) are:
\[\small (4,3)\]
Point \(\small M\) lies:
Therefore:
x-coordinate \(\small =-3\)
y-coordinate \(\small =4\)
Hence, the coordinates of \(\small M\) are:
\[\small (-3,4) \]
Point \(\small L\) lies:
Therefore:
x-coordinate \(\small =-5\)
y-coordinate \(\small =-4\)
Hence, the coordinates of \(\small L\) are:
\[\small (-5,-4) \]
Point \(\small S\) lies:
Therefore:
x-coordinate \(\small =3\)
y-coordinate \(\small =-4\)
Hence, the coordinates of \(\small S\) are:
\[\small (3,-4) \]
| Point | Coordinates | Signs | Quadrant |
|---|---|---|---|
| \(\small B\) | \(\small (4,3)\) | \(\small (+,+)\) | First Quadrant |
| \(\small M\) | \(\small (-3,4)\) | \(\small (-,+)\) | Second Quadrant |
| \(\small L\) | \(\small (-5,-4)\) | \(\small (-,-)\) | Third Quadrant |
| \(\small S\) | \(\small (3,-4)\) | \(\small (+,-)\) | Fourth Quadrant |
Point \(\small A\) lies on the positive x-axis.
Therefore:
x-coordinate \(\small =4\)
y-coordinate \(\small =0\)
Hence, the coordinates of point \(\small A\) are:
\[\small (4,0) \]
Point \(\small B\) lies on the positive y-axis.
Therefore:
x-coordinate \(\small =0\)
y-coordinate \(\small =3\)
Hence, the coordinates of point \(\small B\) are:
\[\small (0,3) \]
Point \(\small C\) lies on the negative x-axis.
Therefore:
x-coordinate \(\small =-5\)
y-coordinate \(\small =0\)
Hence, the coordinates of point \(\small C\) are:
\[\small (-5,0) \]
Point \(\small D\) lies on the negative y-axis.
Therefore:
x-coordinate \(\small =0\)
y-coordinate \(\small =-4\)
Hence, the coordinates of point \(\small D\) are:
\[\small (0,-4) \]
Point \(\small E\) lies on the positive x-axis.
Therefore:
x-coordinate: \[\small \frac{2}{3} \]
y-coordinate: \[\small 0 \]
Hence, the coordinates of point \(\small E\) are:
\[\small \left(\frac{2}{3},0\right) \]
| Point | Coordinates | Position |
|---|---|---|
| \(\small A\) | \(\small (4,0)\) | Positive x-axis |
| \(\small B\) | \(\small (0,3)\) | Positive y-axis |
| \(\small C\) | \(\small (-5,0)\) | Negative x-axis |
| \(\small D\) | \(\small (0,-4)\) | Negative y-axis |
| \(\small E\) | \[\small \left(\frac{2}{3},0\right) \] | Positive x-axis |
Plot the following points on the Cartesian plane:
\[\small A(4,2),\quad B(-5,3),\quad C(-4,-5),\quad D(5,2) \]
Since both coordinates are positive, point \(\small A\) lies in the: First Quadrant.
Since x-coordinate is negative and y-coordinate is positive, point \(\small B\) lies in the: Second Quadrant.
Since both coordinates are negative, point \(\small C\) lies in the: Third Quadrant.
Since both coordinates are positive, point \(\small D\) lies in the: First Quadrant.
The points \[\small A(3,6),\quad B(3,2),\quad C(8,2) \] are the vertices of a rectangle. Plot these points on the Cartesian plane and use the graph to find the coordinates of the fourth vertex \(\small D\).
Draw two mutually perpendicular lines intersecting at the origin \(\small O(0,0)\).
Mark equal divisions on both axes.
Join:
These form two adjacent sides of the rectangle.
Observe carefully:
Hence, the coordinates of point \(\small D\) are:
\[\small D(8,6)\]Plot point:
\[\small D(8,6)\]Join:
Rectangle \(\small ABCD\) is obtained.
| Vertex | Coordinates | Observation |
|---|---|---|
| \(\small A\) | \(\small (3,6)\) | Upper-left corner |
| \(\small B\) | \(\small (3,2)\) | Same x-coordinate as \(\small A\) |
| \(\small C\) | \(\small (8,2)\) | Same y-coordinate as \(\small B\) |
| \(\small D\) | \(\small (8,6)\) | Same x-coordinate as \(\small C\) and same y-coordinate as \(\small A\) |
In rectangles parallel to coordinate axes:
Therefore:
\[\small D=(x\text{-coordinate of }C,\ y\text{-coordinate of }A)\]
\[\small D=(8,6) \]
By plotting the following points on the same graph paper, determine whether the given points are collinear or not:
(i) \[\small (3,5),\quad (1,1),\quad (0,-1) \]
(ii) \[\small (-2,-1),\quad (-1,-4),\quad (-4,1) \]Three or more points are said to be collinear if they lie on the same straight line.
If the points do not lie on a single straight line, they are called non-collinear points.
Draw the x-axis and y-axis perpendicular to each other.
Choose a suitable scale so that all coordinates can be plotted easily.
Plot all the points carefully by:
Plot: \[\small (3,5),\quad (1,1),\quad (0,-1) \]
Plot: \[\small (-2,-1),\quad (-1,-4),\quad (-4,1) \]
Draw straight lines through the plotted points in each group.
Observe whether all points lie on the same straight line.
Points:
\[\small (3,5),\quad (1,1),\quad (0,-1)\]
After plotting and joining these points, we observe that all three points lie on the same straight line.
Therefore, the points are: Collinear
Points:
\[\small (-2,-1),\quad (-1,-4),\quad (-4,1) \]
After plotting and joining these points, one point does not lie on the same straight line formed by the other two points.
Therefore, the points are: Non-collinear
The following concepts form the core foundation of Coordinate Geometry. These points are extremely important for CBSE board examinations, objective questions, viva questions, and higher mathematics.
| Position of Point | Coordinate Form | Observation |
|---|---|---|
| Point on x-axis | \[\small (x,0) \] | y-coordinate is always zero |
| Point on y-axis | \[\small (0,y) \] | x-coordinate is always zero |
| Origin | \[\small (0,0) \] | Both coordinates are zero |
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