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Find the minimum distance from any lattice point to the line $y = \frac{5}{3} x + \frac{12}{7}.$

 Dec 22, 2023
 #1
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(5/3)x  - 1y + 12/7   = 0 

 

5/3  is a  terminating decimal   while   12/7  is  not......so   it will not pass through any lattice points for any x

 

It appears that the closest lattice point  is  (-1,0)

 

See here :  https://www.desmos.com/calculator/mrjzorc1il

 

The distance is

 

l  (5/3)(-1) - 1(0) + 12/7 l               (1/21)

______________________    ≈  _________  ≈ .025

  sqrt ( (5/3)^2 + (-1)^2 )              sqrt (3.56)

 

 

 

cool cool cool

 Dec 22, 2023

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