Question

Find the point on the line -3 x + y - 4 =0 which is closest to the point ( -5, 0 ).

Answer 1

find slope of original line
y = 3 x + 4
so slope = m = 3
so slope of line perpendicular to original = m' = -1/m = -1/3
so find line through (-5,0) with slope m' of -1/3
y = -x/3 + b
0 = 5/3 + b
b = -5/3
so y = -x/3 -5/3
3y= -x - 5 or x + 3y + 5 = 0
find intersection of lines
-9x + 3y -12 = 0
+1x + 3y + 5 = 0
------------------
8 x -17 = 0
x = 17/8
find y
then find the difference in distance between the two points
d^2 = change in x^2 + change in y^2