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Coordinate geometry: Distance formula, Section Formula, Solved Questions and Worksheets

Coordinate geometry is very useful to precisely locate any particle in two or three dimensional space.

We all use Google map.when we have to search for a place. Have you ever thought how Google gets to know the exact location of that place.The answer to this question is very simple google use the coordinates of that place to exactly navigate to that place.

Coordinates of a point in Two dimension

Cartesian coordinate system is widely used to locate a particle in two dimensional or Three dimensional Plane.
In Cartesian coordinate system the coordinate of orgin are defined as (0,0). Origin is always taken as reference point to get the coordinates of any other point in this system.

The notation to write the coordinates of a point P is (x,y). Where x is the distance from origin in x direction and y is the distance from origin in y direction.

The Cartesian coordinate system is Divided into four parts. i.e four quadrants

1 st Quadrant

In the first quadrant, For every point the value of x and y is always Postive.

Example: (3,1) ,(7,8) ,(3,7),(3,0) all these points lie in first quadrant

General Notation (x,y) = (+,+)

2nd Quadrant

In the second quadrant, For every point the value of x is Negative and for  y is  Postive.

Example: (-2,1) ,(-7,0) ,(-3,5),(-1,9) all these points lie in second quadrant.

General Notation (x,y) = (-,+)

3rd Quadrant

In the third quadrant, For every point the value of x and y are Negative.

Example : (-2,-2) ,(-2,-5) ,(-7,-2),(-2,-8) all these points lie in third quadrant.

General Notation (x,y) = (-,-)

4th Quadrant

In the 4th  quadrant, For every point the value of x is positive and for y is Negative.

Example : (1,-2) ,(5,-5) ,(6,-7),(6,-1) all these points lie in third quadrant.

General Notation (x,y) = (+,-)

For instance if a point A has coordinates (3,4) which means it is 3 unit away  from origin in +x-axis and 4 unit away from origin in +Y-axis.

Distance Formula 

The distance between two points A(x1,x2) and B(x2,y2) is given by the following formula.

AB = √(x2-x1)^2 +(y2-y1)^2

Example 1 : The coordinates of the points P and Q are (3,2) and (8,5). Find the Distance between PQ.

Solution: x1 = 3 ,   y1=2
x2 = 8 ,   y2=5

PQ = √(8-3)^2 + (5-2)^2
      = √ 25+9
      = √34

Example 2 : The distance between two points is 5 and Coordinates of point M and N are (2,4) and (5,x).

Solution: 5 = √(5-2)^2 + (x-4)^2

25 = 9 + (x-4)^2

√16 = x-4
x = 4+4 
   = 8

Section Formula 

The coordinates of a point that divides the line PQ into m:n is :

X = (mx 1+n x 2)/m+n

Y =  (my 1+n y 2)/m+n


If the Point divides the line PQ in 1:1 then the point must be mid point and its coordinates are given by :

X = (x1+x2)/2.  

Y = (y1+y2)/2

Example 1 : Two points A(7,8) and B(5,2) form a line.find the coordinates of the the point that divides it into 4:2.

Ans : Let the coordinates of that point be (X,Y)

X = (4×5+2×7)/(4+2)
X= 34/6

Y = ( 4×2 + 2×8)/(4+2)
   = 4

Hence the coordinates of the point that divides the line joining A(7,8) and B(5,2) is:
(34/6,4)

Outer Section formula 

When the dividing point is outside the line in that case 
Let the coordinates of that point be (X,Y) then 

X= (mx 2-nx 1)/(m-n)

Y =  (my 2-ny 1)/(m-n)

The coordinates of that point are:

C(X,Y) = ((mx 2-nx 1)/(m-n),(my 2-ny 1)/(m-n))

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