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Co-oridante Axes and Representation of a Point

The figure given alongwith is called the X-Y Cartesian plane. The line XOX′ is called the X-axis and YOY′ the Y-axis.
 
Description: 15102.png
 
If P(xy) is a point in this plane, then x is the X-coordinate of P or abscissa of P and y is called the Y-coordinate of P or the ordinate of P.
 
Remember that X-coordinate of the point is the distance of the point from Y-axis and Y-coordinate of the point is the distance of the point from X-axis.
 
The X-Y Cartesian plane is divided into four equal parts called quadrants (I, II, III, IV).
 
Sign convention
 

 

1st
Quadrant

2nd
Quadrant

3rd
Quadrant

4th
Quadrant

X-axis

+ve

-ve

-ve

+ve

Y-axis

+ve

+ve

-ve

-ve

Equation and Graph of Co-ordinate Axis

  1. Equation of X and Y-axis are Y = 0 and x = 0 respectively
     
    Description: 15117.png
  2. Equation of a line parallel to X-axis is Y = b (b is constant)
     
    Description: 15130.png
  3. Equation of a line parallel to Y-axis is X = a (a is constant)
     
    Description: 15138.png
  4. Any point on the X-axis can be taken as (a, 0) and any point on the Y-axis can be taken as (0, b).
     
    Description: 15146.png
  5. To find out X and Y intercepts of a line, we will put Y = 0 and X = 0 respectively in the equation of the line.
If you know the co-ordinates of two points:
  • Find out the distance between them.
  • ​Find the mid-point, slope and equation of the line segment formed by these two points.

Distance between two points

If there are two points A(x1y1) and B(x2y2) on the XY plane, then the distance between them is given by Description: 13968.png
 
Example-1
What is the distance between the points (3, 2) and (6, 6)?
Solution
Distance = Description: 13975.png = Description: 13955.png = 5 units.
 
 
Example-2
Coordinates A and C of a square ABCD (points in order) are (4, 2) and (1, 4). What is the area of the square?
Solution
For square ABCD, line segment AC will be its diagonal.
 
AC = Description: 13982.png = Description: 13993.png units
 
Diagonal of square = Description: 14003.png = Description: 14020.png units
 
So, side of square = Description: 14027.png units
 
Hence, area = (Side of square)2 = Description: 14040.png = 6.5 sq. units
 

 

Methods to identify if three points A, B and C are in the same straight line:

If there are three points A, B and C, they may be in the same straight line or form a triangle.

Method 1:

 

Area formed by the three points = 0 [Formula to find out the area of triangle given ahead]

Method 2:

 

Slope of any two line segment AB or BC or AC are equal. For example, slope of line AB = Slope of line AC

Method 3:

 

Sum of any two line segment is equal to the equal to third line segment. For example, AB + BC = AC
 

Division of a line segment [if three points A, B and C are in a straight line]

  1. Internal
     
    The coordinates of a point P which divides the line joining A(x1, y1) and B(x2y2) internally in a ratio l:m are given by
     
    Description: 14047.png
  2. External
     
    The coordinates of a point P which divides the line joining the point A(x1, y1) and B(x2y2) externally in the ratio l:m are given by
     
    Description: 14067.png

The image of a point along the mirror place on a straight line

The image of A(x1, y1) with respect to the line mirror ax + by + c = 0 be B(x2y2) is given by

Description: 14087.png 

Description: 14078.png
 
Description: 15264.png

Foot of the perpendicular

If the foot of the perpendicular from (x1y1) to the line lx + my + n = 0 is (hk) then Description: 14097.png





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