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.

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

Terms Used in Analytic/Coordinate Geometry

Let us learn the terminology used in analytic geometry, such as;

  • Plane
  • Coordinates

What is a Plane?

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.

What are Coordinates?

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
1
2 x
3

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.

What are Cartesian Coordinates?

The most well-known coordinate system is the Cartesian coordinate to use, where every point has an x-coordinate and y-coordinate expressing its horizontal position, and vertical position respectively. They are usually addressed as an ordered pair and denoted as (xy). We can also use this system for three-dimensional geometry, where every point is represented by an ordered triple of coordinates (xyz) in Euclidean space.

What are Polar Coordinates?

In case of polar coordinates, each point in a plane is denoted by the distance ‘r’ from the origin and the angle θ from the polar axis.

What are Cylindrical Coordinates?

In case of cylindrical coordinates, all the points are represented by their height, radius from z-axis and the angle projected on the xy-plane with respect to the horizontal axis. The height, radius and the angle are denoted by h, r and θ, respectively.

What are Spherical Coordinates?

In spherical coordinates, the point in space is denoted by its distance from the origin ( ρ), the angle projected on the xy-plane with respect to the horizontal axis (θ), and another angle with respect to the z-axis (φ).

What is 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.
Coordinate Plane in Analytic Geometry
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.

Let’s learn more about Co-ordinates:

  • 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 coordinate Geometry. Few of the important ones are being used to find Distance, Slope or to find the equation of the line.

Distance Formula

Let the two points be A and B, having coordinates to be (x1,y1) and (x2,y2) respectively.

Thus, the distance between two points is given as-

d = √[(x2-x1)2+(y2-y1)2]

Midpoint Theorem Formula

Let A and B are some points in a plane, which is joined to form a line, having coordinates (x1,y1) and (x2,y2), respectively. Suppose, M(x,y) is the midpoint of the line connecting the point A and B then its formul is given by;

M(x,y) = [(x1+x2/2),(y1+y2/2)]

Angle Formula

Let two lines have slope m1 and m2 and θ is the angle formed between the two lines A and B, which is represented as;

tan θ = m1-m2/1+m1m2

Section Formula

Let two lines A and B have coordinates (x1,y1) and (x2,y2), respectively. A point P the two lines in the ratio of m:n, then the coordinates of P is given by;

Analytic Geometry-Section Formula

 

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.

Let’s see how three-dimensional number space is represented on a geometric space.

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.

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