Point Coordinates in a Plane :: Origin, Points on Axes; Points on Lines Parallel to Axes

Learning Analytical/Coordinate/Cartesian Geometry
...from page 9

Coordinates of Points in a Plane

Coordinates of the Origin

The origin (Point "O") lies on both the axes. The coordinate of the origin with respect to the horizontal axis is O(0) and with respect to the vertical axis is also O(0).

The projection of a point on a line to which it belongs is the point itself.
⇒ The projection of "O" on

x-axis is "O(0)"
y-axis is "O(0)"

The Coordinates of the origin in the plane ⇒ O = (0,0)

Distance of a point on the axes from the axes

Points on the x-axis
- Considering "A" and "H" to be on the Horizontal Coordinate Line
  
  Distance of a point
  - From the origin is given by the magnitude of the coordinate itself.
    
    Eg: (a) A = − 4 ⇒ OA = 4 units
    
    (b) H = 3 ⇒ OH = 3 units
  Projection of a point
  - On the Coordinate Line is the Point itself [Since the point lies on the line.]
    
    Eg: (a) Projection of "A" on x'Ox is "A"
    
    (b) Projection of "H" on x'Ox is "H"
- Considering "A" and "H" to be in the Coordinate plane
  
  Distance of a point
  - From the origin is given by the magnitude of the x-coordinate of the point.
    
    Eg: (a) A = (−4, 0) ⇒ Distance of "A" from "O" i.e. OA is 4 units
    
    (b) H = (3, 0) ⇒ Distance of "H" from "O" i.e. OH is 3 units
  - From the y-axis is equal to the distance of the point from the origin i.e. the magnitude of the x-coordinate of the point.
    
    Eg: (a) A = (−4, 0) ⇒ Distance of "A" from the y-axis = OA = 4 units
    
    (b) H = (3, 0) ⇒ Distance of "H" from the y-axis = OH = 3 units
Points on the y-axis
- Considering "P" and "Q" to be on the Vertical Coordinate Line
  
  Distance of a point
  - From the origin is given by the magnitude of the coordinate itself.
    
    Eg: (a) P = 2 ⇒ OP = 2 units
    
    (b) Q = − 2 ⇒ OQ = 2 units
  Projection of a point
  - On the Coordinate Line is the Point itself [Since the point lies on the line.]
    
    Eg: (a) Projection of "P" on yOy' is "P"
    
    (b) Projection of "Q" on yOy' is "Q"
- Considering "P" and "Q" to be in the Coordinate plane
  
  Distance of a point
  - From the origin is given by the magnitude of the y-coordinate of the point.
    
    Eg: (a) P = (0, 2) ⇒ Distance of "P" from "O" i.e. OP is 2 units
    
    (b) Q = (0, − 2) ⇒ Distance of "Q" from "O" i.e. OQ is 2 units
  - From the x-axis is equal to the distance of the point from the origin i.e. the magnitude of the y-coordinate of the point.
    
    Eg: (a) P = (0, 2) ⇒ Distance of "P" from the x-axis = OP = 2 units
    
    (b) Q = (0, − 2) ⇒ Distance of "Q" from the x-axis = OQ = 2 units

Coordinates of Points belonging to the Coordinate axes

Points on the x-axis
The First Coordinate (x-coordinate) is
- The projection of the point on the x-axis.
  (Or)
- The distance of the point from the y-axis (Or) the origin.
The second Coordinate (y-coordinate) is
- Zero
  [A(x_a, y_a) ∈ x – axis ⇒ y_a= 0]
Points on the y-axis
The First Coordinate (x-coordinate) is
- Zero
  [P(x_p, y_p) ∈ y – axis ⇒ x_p= 0]
The second Coordinate (y-coordinate) is
- The projection of the point on the y-axis.
  (Or)
- The distance of the point from the x-axis (Or) the origin.

Points on the x-axis would be of the form (x, 0) and on the y-axis would be of the form (0, y).

Coordinates of Points on lines parallel to the Coordinate axes

If a line is parallel to the coordinate axes, then all the points on the line would have the same distance from the coordinate axis to which it is parallel.

Lines Parallel to the x-axis

For all the points on a line parallel to the x-axis
- The distance from the x-axis would be the same.
- The y-coordinate of a point is its distance from the x-axis.
M (x_m, y_m), N(x_n, y_n) are points in the same plane. || x-axis ⇔ y_m = y_n.
Lines Parallel to the y-axis

For all the points on a line parallel to the y-axis
- The distance from the y-axis would be the same.
- The x-coordinate of a point is its distance from the y-axis.
A (x_a, y_a), C(x_c, y_c) are points in the same plane. || y-axis ⇔ x_a = x_c.

Corollary

If two points on a line have the same x-coordinate, then the line is parallel to y-axis

If two points on a line have the same y-coordinate, then the line is parallel to x-axis

General Form of representing a point in a plane

Using Number Subscripts
A point in a plane in its general form is represented as (x₁, y₁) with the subscripts indicating the sequence in which the points are being considered. (x₁, y₁) indicate Coordinates of the First point, (x₂, y₂) indicate the coordinates of the second point etc., the point itself being represented by the capital letters of the english alphabets.
Eg: P = (x₁, y₁) ; A (x₁, y₁) ; M = (x₂, y₂) etc.
Using alphabet Subscripts
Points can also be represented using the lower case letters (of the letter used to represent the point) for subscripts.
Eg: P = (x_p, y_p) ; A (x_a, y_a) ; M = (x_m, y_m) etc.

We find the alphabet subscripts more convenient at some places and the letter subscripts at the others. It is the students choice to decide on what he/she wants to use. We use the letter subscripts as we find it easy to associate the coordinates with the points.

Sign of the Coordinates

The coordinates are numerical values with signs (+ or −) placed before them based on the location of the point they are representing. When the sign is + it is a convention that we ignore placing the sign. Therefore where there is no sign for the coordinate we assume it to be +ve.

Author Credit: The Edifier
...continued page 11

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