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Im searching for an equation that governts the movement of a point such that the sum of its distances from the coordinate axes is minimal.....
for convinience sake just consider the first quadrant.... plz give an elaborate answer.... im abit simple in maths... :rolleyes:
This is what I could come up with...Quote:
Originally Posted by The Godfather
Let y=f(x) the equation we're after. Then the sum of distances to be minimized is:
x2 + y2 = x2 + f(x)2
If it's to be minimized, then the derivative must be 0:
0 = d/dx(x2 + f(x)2) = 2x + 2f(x)*f'(x)
or
f'(x) = -x/f(x)
which for convenience can be written as
dy/dx = -x/y
and rearranged:
ydy = -xdx
and integrated to yield:
y2/2 = -x2/2 + C
C being an integration constant which, being arbitrary, can be written as
y2/2 = -x2/2 + k2/2
without loss of generality. After rearrangement this finally gives:
y = Sqrt(k2 - x2)
HTH
I was about to say that myself.
I think my post above is not correct... What I was trying to minimize in it was not the sum of distances to the x and y axes but the distance to the origin x2 + y2
so that's why I got the equation of a circle of radius k.
Actually, the sum of distances to be minimized is:
x + y = x + f(x)
Taking the derivatives:
0 = d/dx(x + f(x)) = 1 + f'(x)
or
f'(x) = dy/dx = -1
or
dy = -dx
so that, finally:
y = C - x
the equation of a straight line, with C being the integration constant.