Find the point on the line 4x+2y+5=0 which is closest to the point (-2,-1)

Answer 1

without calculus, the distance from the point to the line is

|4(-2)+2(-1)+5|/√(4^2+2^2) = √5/2

Using calculus, we have the distance z from (-2,-1) to (x,y) is

z^2 = (x+2)^2 + (y+1)^2
But y = -(4x+5)/2, so
z^2 = (x+2)^2 + ((-(4x+5)/2)+1)^2
z^2 = 5x^2 + 10x + 25/4

z = √(5x^2 + 10x + 25/4)
dz/dx = 10(x+1)/√(5x^2+10x+25/4)

dz/dx=0 at x=-1
z(-1) = √5/2