Welcome to the first lesson in Coordinate Geometry Basics!
Coordinate geometry is an interesting topic to study and explore, and one of my favorites too.
There are usually multiple ways of solving a given problem in this topic, which makes it quite interesting. I’ll try to illustrate and compare different methods of solving a given problem, wherever applicable. This will help you understand the best ways to approach and solve a problem.
This topic can be divided into five sub-topics – Straight Line, Circle, Parabola, Ellipse and Hyperbola, each of which I’ll cover individually. But before diving into these sub-topics, we must first understand the basics of coordinate geometry.
Lets get started !
Coordinate geometry is the study of geometry using algebra, that is using relations, operations, equations etc. to represent geometrical figures and establish various results, properties, proofs etc. about them. This will make more sense as we proceed.
To translate a problem in geometry into algebraic form, we use something known as a ‘coordinate system’ – a system which uses numbers (coordinates) to uniquely determine the location of a point in a plane (or space).
The Cartesian Coordinate System
We’ll be particularly using the Cartesian coordinate system – a particular type (among many) which represents each point by using a pair of number (coordinates) which are the signed distances of the point from two fixed perpendicular lines, known as the axes.
Here’s what the Cartesian coordinate system looks like:
With reference to the above figure, here are some terminologies, notations and conventions.
- The horizontal line is referred to as the X axis, and the vertical as Y axis. The axes divide the plane into four quadrants (labelled in red)
- The point of intersection of the two axes is referred to as the Origin (O)
- The coordinates of each point are denoted by an ordered pair of numbers (x, y)
- The x-coordinate of a point is referred to as its abscissa and the y-coordinate as its ordinate
- The abscissa of a point is its ‘signed’ distance from the Y-axis. By signed, it means that towards the right of the Y-axis, the abscissa is positive, whereas on the left it is negative. (This is a convention).
- Similarly, the ordinate of a point is its signed distance from the X-axis.
- Using the above convention, the origin has the coordinates (0, 0) and we can determine the signs of x and y coordinates of a point (indicated in blue) in the four quadrants
Here is another figure to illustrate the above points.