Analytic Geometry

Analytic geometry is that branch of geometry in which the position of the point on the plane can be located using an ordered pair of numbers called as Coordinates. This is also called coordinate geometry or cartesian geometry. This geometry is a contradiction to the synthetic geometry, where there is no use of coordinates or formulas. It is considered axiom or assumptions, to solve the problems.

Analytic geometry has its use in both two dimensional and three-dimensional geometry. It is used to represent geometrical shapes.

Analytic Geometry Terms

  • Plane
  • Coordinates


To understand how analytic geometry is important and useful, First, We need to learn what a plane is?

If a flat surface goes on infinitely in both the directions, it is called a Plane.

So, if you find any point on this plane, it is easy to locate it using Analytic Geometry. You just need to know the coordinates of the point in X and Y plane.


Coordinates are the two ordered pair, which defines the location of any given point in a plane. Let’s understand it with the help of the box below.

  A B C
2   x  

In the above grid, The columns are labelled as A, B, C, and the rows are labelled as 1, 2, 3.

The location of letter x is B2 i.e. Column B and row 2. So, B and 2 are the coordinates of this box, x.

As there are several boxes in every column and rows, but only one box has the point x, and we can find its location by locating the intersection of row and column of that box.

The Coordinate Plane

In coordinate geometry, every point is said to be located on the coordinate plane or cartesian plane only.

Look at the figure below.

The above graph has x-axis and y-axis as it’s Scale. The x-axis is running across the plane and Y-axis is running at the right angle to the x-axis. It is similar to the box explained above.

  • Origin:

It is the point of intersection of the axis(x-axis and y-axis). Both x and y-axis are zero at this point.

  • Values of the different sides of the axis:

x-axis – The values at the right-hand side of this axis are positive and those on the left-hand side are negative.

y-axis – The values above the origin are positive and below the origin are negative.

  • To locate a point

We need two numbers to locate a plane in the order of writing the location of X-axis first and Y-axis next. Both will tell the single and unique position on the plane. You need to compulsorily follow the order of the points on the plane i.e The x coordinate is always the first one from the pair. (x,y).

If you look at the figure above, point A has a value 3 on the x-axis and value 2 on the Y-axis. These are the rectangular coordinates of Point A represented as (3,2).

Analytic Geometry Formulas

Graphs and coordinates are used to find measurements of geometric figures. There are many important formulas in Analytic Geometry. Few of the important ones are being used to find Distance, Slope or to find the equation of the line. Check out coordinate geometry formulas to know all of the above-mentioned formulas in a detailed way.

Analytic Geometry of Three Dimensions

In this, we consider triples (a,b,c) which are real numbers and call this set as three- dimensional number space and denote it by R’. All the elements in the triple are called coordinates.

In three-dimensional space, we consider three mutually perpendicular lines intersecting in a point O. these lines are designated coordinate axes, starting from 0, and identical number scales are set up on each of them.

Analytic Geometry Applications

Analytical geometry has made many things possible like the following:

  • We can find whether the given lines are perpendicular or parallel.
  • We can determine the mid-point, equation, and slope of the line segment.
  • We can find the distance between the points.
  • We can also determine the perimeter or the area of the polygon formed by the points on the plane.
  • Define the equations of ellipse, curves, and circles.

Example Problem

Question: What is the point of intersection of the axis (X-axis and Y-axis) called?

Answer: The point of intersection of the axis (X-axis and Y-axis) called Origin and X and the Y-axis is 0 at this point.