Find point of the line

Question

Find the point on the line -5 x + 2 y + 3 =0 which is closest to the point ( -5, -3 ).

Answer 1

The distance from any point (x,y) to the point (-5,-3) is given by the distance formula 

 

d = sqrt ( (x - (-5))² + (y - (-3))² ) = sqrt ( (x+5)² + (y+3)² )  

 

d will be minimized when g = d² = (x+5)² + (y+3)² is minimized 

 

If the point (x,y) is on the line -5x + 2y + 3 = 0 then 

2y = 5x - 3  ==>  y = (5/2)x - (3/2) so 

 

g(x) = (x+5)² + ((5/2)x +(3/2))²  

g'(x) = 2(x+5) + 2((5/2)x + (3/2))(5/2)

        = 2x + 10 + 5((5/2)x + (3/2)) 

        = 2x + 10 + (25/2)x + (15/2)

        = (29/2)x + (35/2)  

 

Note that g''(x) = (29/2) > 0 so any critical point will be a minimum 

 

g'(x) = 0  ==>  (29/2)x + (35/2) = 0

                        29x = -35

                        x = -35/29 

 

So y = (5/2)x - (3/2) = (5/2)(-35/29) - (3/2) = -175/58 - 87/58 = -262/58 = -131/29  

 

So the point is (-35/29,-131/29) 

 

You can check the answer by noting that the slope of the line from 

(-35/29,-131/29) to (-5,-3) is 

   (-3 - (-131/29)) / (-5 - (-35/29)) 

= ((-87 + 131)/29) / ((-145 + 35)/29) 

= 44 / (-110) 

= -2/5 

 

which is the negative reciprocal of the slope of the given line, which means that the 2 points lie on a line that is perpendicular to the given line, which means that these 2 points form the shortest distance between the point and the line