Coordinate Line :: One Point : Location; Two points : Relative location, distance.

Learning Analytical/Coordinate/Cartesian Geometry
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Coordinate Line » Properties

The properties of a coordinate line are true whatever may be its position. They hold good for horizontal, vertical and oblique lines. To aid easier understanding, we have explained the properties using a "Horizontal Line" and therefore, used the terms to the left of the origin and to the right of the origin. They would mean to the negative side and to the positive side respectively, when we consider any line.

The properties are:

For every real number ‘x’ and for each point ‘P’ on l a one to one correspondence exists.
P(x) denotes the point ‘P’ with coordinate ‘x’, Q (y) denotes the point ‘Q’ with coordinate ‘y’.
Location of a point
Where P(x) ∈ l
- The coordinate is zero ⇒ The point lies at the origin.
  [For P(x), x = 0 ⇒ ‘P’ lies at the origin i.e. ‘P’ and ‘O’ coincide.]
- The coordinate is positive ⇒ The point lies towards the right of the origin.
  [For P(x), x > 0 ⇒ ‘P’ lies to the right of ‘O (0)’.]
- The coordinate is negative ⇒ The point lies towards the left of the origin.
  [For P(x), x < 0 ⇒ ‘P’ lies to the left of ‘O (0)’.]
Relative Location of two points
Numerical Value ⇒ Magnitude and sign of a number taken together.
Between two points the point with a higher numerical value of the coordinate is to the right.
Where P (x), Q (y) ∈ l
- P (x) is to the right of Q (y) where x > y.
  [For P(x), Q (y), x > y 0 ⇒ ‘P’ lies to the right of ‘Q’.]
  In deciding the relative positions of the two points, the location of the origin is irrelevant.
  The points may be either on
  - Either side of the origin or
    
    Eg: P (2) and Q(− 3).
    
    2 > − 3 ⇒ P is to the right of Q.
  - The Same side of the origin.
    
    Both Negative Side
    
    Eg: P (− 2) and Q(− 5).
    
    − 2 > − 5 ⇒ P is to the right of Q.
    
    Both Positive Side
    
    Eg: P (2) and Q(4).
    
    4 > − 2 ⇒ P is to the right of Q.
  Corollary:
  P (x) is to the left of Q (y) where x < y
  [For P(x), Q (y), y > x 0 ⇒ ‘Q’ lies to the left of ‘P’.]
- P (x) coincides with Q (y) where x = y
  [For P(x), Q (y), x = y 0 ⇒ ‘P’ coincides with ‘Q (y)’.]
  
  Eg: P (2) and Q(2).
  
  2 = 2 ⇒ P and Q coincide.
Distance Between Two Points
Distance Between Two Points on the coordinate line is given by the difference of the numerical values of their coordinates.
Where A (x_a), B (x_b) ∈ l

Distance between A and B = Numerical Value of A − Numerical Value of B

Distance between A and B = x_a − x_b

This calculation would give either a positive value or a negative value based on from which side we started measuring the distance.
The resultant is negative ⇒ The first point (A) is to the left of the second (B).
The resultant is positive ⇒ The first point (A) is to the right of the second (B).
Since we can identify the location of the point by comparing the numerical values of the coordinates of the points and we need only to calculate the distance between the two points, we use the modulus value of the difference to derive the required answer.
Therefore,

Distance between A and B = | x_a − x_b |

It is also interpreted as the Length of the line segment joining any two points on the coordinate line
Distance of a point on the coordinate line from the origin
It is the length of the line segment joining the origin and the point on the coordinate line. It is given by
Distance between the Origin and A (x_a) ⇒ AO = | x_a |
Horizontal Line

Eg: A (− 4) and B (+ 3).

Distance between A and B = |(x_A) − (x_B)|

= |(− 4) − (+ 3)|

= |− 4 − 3|

= |− 7|

⇒ AB = 7 units

Distance of A from the Origin = | (x_A) |

= |(− 4)|

= |− 4|

⇒ OA = 4 units

Vertical Line

Eg: M (3) and N (− 2).

Distance between M and N = |(x_M) − (x_N)|

= |(3) − (− 2)|

= |3 + 2|

= |5|

⇒ MN = 5 units

Distance of N from the Origin = |(x_N)|

= |− 2|

⇒ ON = 2 units