Foot of perpendicular short trick

 Foot of Perpendicular from a given point on a line

Let  P : (x1 ,  y1) be the point and L : (ax + by + c = 0) be the line .

Let (x , y) be the foot of perpendicular. 

Method 1 :

Let  '' m1 ''  be the slope of  line '' L ''.

m1 = -a / b

Now , we make a line perpendicular to  '' L '' passing through ''  P  ''.

So, slope of that line is (-1 / m1) and passing point point is (x1 , y1).

We can find line equation by using point slope form.

And, then find the intersection point of given line and above line.

But this method is time consuming.

Method 2 : 

Example :

Let 2x + y = 3 be the line and (1, 2) be the point .

So,

L :  2x + y - 3 = 0   { a = 2  ;   b = 1 ;  c = -3}

P : (1 ,  2) {x1 = 1 ;  y1  = 2}

Now, 

x - 1 = y - 2 =  -1

    2           1           5  

  x - 1   =   -1       

    2              5

and

 y - 2 =  -1

    1           5

On solving:

x = 3/5

y = 9/5

So, foot of perpendicular (3/5 , 9/5).