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Math Help - Minimise Radius- With a Nasty Twist.

  1. #1
    Junior Member qspeechc's Avatar
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    Minimise Radius- With a Nasty Twist.

    Q: A lattice point in the plane is a point with integer co-ordinates. Suppose that circles with radius r have lattice points as centres. Find the smallest r such that any line with gradient 2/5 intersects some of these circles.

    That is the exact phrasing of the question. By "some" I guess they mean at least two.
    Below is the answer

    A: sqrt(29)/58
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  2. #2
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    Opalg's Avatar
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    I agree with the answer 1/(2√29) (=√(29)/58).

    To see this, you need to know that the distance d from a point (m,n) to a line ax+by+c=0 is given by the formula d = \frac{|am+bn+c|}{\sqrt{a^2+b^2}}.

    Look at the line 2x-5y=0. This passes through the lattice point (5k,2k) for any integer k. Apart from these points, the lattice point that come closest to this line are those like (2,1) and (3,1) (as you can easily see by drawing a diagram). Their distance from the line is given by the above formula as \frac1{\sqrt{5^2+2^2}} = \frac1{\sqrt{29}}.

    Now think what happens when this line is shifted to a parallel line. So long as the distance shifted (meaning the perpendicular distance from the original line) is less than \frac1{2\sqrt{29}}, the shifted line will still be within that distance of the origin. But as soon as the distance shifted is greater than that, the shifted line will be within that distance of (2,1) (if it is being shifted upwards) or (3,1) (if it is being shifted downwards).

    The same reasoning applies to any other line with gradient 2/5. It will always be within distance 1/(2√29) of (infinitely many) lattice points.
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  3. #3
    Junior Member qspeechc's Avatar
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    Er, thanks, I think I may need a few hours to understand that...
    Do I need to check that (2,1) and (3,1) etc. are the closest points. It's no good saying they look the closest...
    Not quite sure I fully get it, but I guess I just have to go over it a million times first...
    Thank-you, anyhow, this problem was really troubling me
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  4. #4
    MHF Contributor
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    Quote Originally Posted by qspeechc View Post
    Do I need to check that (2,1) and (3,1) etc. are the closest points. It's no good saying they look the closest...
    That's a fair point. What you need to say is that the line passes through the lattice point (5,2), so you only need to check the closest points in the interval 0<x<5. I think it's clear that closest points in this interval are (2,1) and (3,1). In other intervals, the pattern repeats itself. So in the interval 5<x<10 the closest points are (7,3) and (8,3). In the interval 10<x<15 it will be (12,5) and (13,5), and so on. In each interval of length 5 between lattice points that lie on the line, there will be two points at a distance 1/√29 from the line.
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