What is coordinate geometry?
A very important branch of mathematics in which geometric problems are solved through algebra by using the coordinate system is known as coordinate geometry.
Note: If the abscissa of a point is x and the ordinate is y, then (x, y) are called the coordinates of the point.
Cartesian System:
The system used for locating any point on the plane is called the Cartesian system or coordinate system.
In the Cartesian system, there are two perpendicular lines. These two lines to locate a point in a plane. One line will be horizontal, and the other will be vertical. The horizontal line XX’ is called the X-axis, and the vertical line YY’ is called the Y-axis. The point where these lines intersect each other is called the origin and is denoted by ‘O’. OX and OY are called positive directions, while OX’ and OY’ are called negative directions.
The axes (plural of the word ‘axis’) divide the plane into four parts. These four parts are called the quadrants (one fourth part), numbered I, II, III and IV anticlockwise from OX
So, the plane consists of the axes and these quadrants. We call the plane, the Cartesian plane, or the coordinate plane, or the xy-plane. The axes are called the coordinate axes.
Sign Convention with Quadrants:
The ray OX on the X-axis is taken as positive, OX’ as negative X-axis, OY on the Y-axis as positive, and OY’ as negative.
I-Quadrant (+, +): The point is in the 1st quadrant then the 1st quadrant is enclosed by the positive X-axis and the positive Y-axis so the point will be in the form (+, +) e.g. (2,5)
II-Quadrant (-, +): A point is in the 2nd quadrant then the 2nd quadrant is enclosed by the negative X-axis and the positive Y-axis so the point will be in the form (-, +) e.g. (-2,5)
III-Quadrant (-, -)-A point is in the 3rd quadrant then the 3rd quadrant is enclosed by the negative X-axis and the negative Y-axis so the point will be in the form (-, -) e.g. (-2,-5)
IV-Quadrant (+, -)-A point is in the 4th quadrant then the 4th quadrant is enclosed by the positive X-axis and the negative Y-axis so the point will be in the form (+, -) e.g. (2,-5)
| Quadrant | sign of x | sign of y |
|---|---|---|
| I | + | + |
| II | – | + |
| III | – | – |
| IV | + | – |
Some Important Questions on Coordinate Geometry:
Question 1. Plot the point \( (-5, 4) \) and identify its quadrant.
Solution: \( x = -5 \) move 5 units left, \( y = 4 \) move 4 units up. Thus, the point \( (-5, 4) \) lies in the second quadrant.
Question 2. Find the coordinates of a point which is 6 units away from the origin along the positive Y-axis.
Solution: On the positive Y-axis, \( x = 0 \).
Distance = 6 units along the positive y-axis; thus, \( y = 6 \).
\[(0, 6)\]
Question 3. In which quadrant does the point \( (-7, -9) \) lie?
Solution: \( x = -7 \) (negative), \( y = -9 \) (negative).
Thus, \( (-7, -9) \) lies in the third quadrant.
Question 4. Find the coordinates of the point which lies on the X-axis at a distance of 8 units from the origin towards the negative direction.
Solution: On X-axis, \( y = 0 \). Distance towards negative side: \( x = -8 \). Thus, the coordinates are \((-8,0) \).
🔍 Frequently Asked Questions (FAQs) on Coordinate Geometry
Q1. What is the Cartesian system?
Answer: The system used for locating any point on the plane is called the Cartesian system
Q2. How many quadrants are there in the Cartesian plane?
Answer: Four quadrants.
Q3. What are the coordinates of the Origin?
Answer: (0, 0).
