ADVANCED MATHEMATICS FORM 5 – COORDINATE GEOMETRY – part 2

THE SHORTEST/PERPENDICULAR DISTANCE FROM THE POINT TO THE LINE

Introduction:

                           

     From the figure above it be loved that PC is the only one posses shortest distance rather than the other as perpendicular to the line that is way, our intention is to get shortest / perpendicular distance from the point P to the line.

                 

   Our intention is to find shortest distance from point to the line          .

  From

   

   

    

    

   Since

    

   

   

    

From the trigonometric identity

  

 

Then

 

But

   

   

   

  

Also

   

   

Since the point R is on the line

Hence;

  

 

 

 Substitute the value of into 1 above

Then

  

   

  

   

Apply square root both sides

   =

   = +

Hence

  

   

At the point

Note:
The distance from the origin to the line ax +by + c = 0 is given by;

  

Example:

1.        i) Find the perpendicular distance from point for the line

2.      ii) Find the value of K if perpendicular distance from point for the line is units.

3.      iii) Find the shortest distance from the origin to the line

4.      If the shortest distance from the point to the line is 3 units. Find the value fm.

Solution

Given: point

  Line

Recall:

 

   The perpendicular distance Is 4 unit

Solution:

  Given point

    

 

Required k

Then