"Your task is to derive the equation for the distance between a point and a line using the method of Lagrange Multipliers. Use Lagrange Multipliers to establish the formula D=(|ax_0+by_0-d|)/(Sqrt(a^2+b^2)) for the distance D from the point (x_0,y_0) to the line ax+by=d."
Please help. We've tried backwards and frontwards for a very long time, now. Thank you in advance!
Please help. We've tried backwards and frontwards for a very long time, now. Thank you in advance!
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Let (x, y) be a point on ax+by = d.
f(x, y) = distance^2 = (x-x0)^2 + (y-y0)^2 + λ(ax+by-d)
f'x = 2(x-x0) + λa = 0 => x= x0 - λa/2
f'y = 2(y-y0) + λb = 0 => y = y0 - λb/2
ax+by = d becomes a(x0 - λa/2) + b(y0 - λb/2) = d
λ = 2(ax0 + by0 - d)/(a^2+b^2)
So,
distance, min
= sqrt[ (x-x0)^2 + (y-y0)^2]
= sqrt[λ^2(a^2 + b^2)/4]
= |λ/2|sqrt(a^2+b^2)
= |ax0+by0-d|/sqrt(a^2+b^2)
QED
f(x, y) = distance^2 = (x-x0)^2 + (y-y0)^2 + λ(ax+by-d)
f'x = 2(x-x0) + λa = 0 => x= x0 - λa/2
f'y = 2(y-y0) + λb = 0 => y = y0 - λb/2
ax+by = d becomes a(x0 - λa/2) + b(y0 - λb/2) = d
λ = 2(ax0 + by0 - d)/(a^2+b^2)
So,
distance, min
= sqrt[ (x-x0)^2 + (y-y0)^2]
= sqrt[λ^2(a^2 + b^2)/4]
= |λ/2|sqrt(a^2+b^2)
= |ax0+by0-d|/sqrt(a^2+b^2)
QED