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| The Day of the Locus (Posted on 2006-12-07) |
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What is the locus of points that are exactly n times farther from P2 than from P1?
(P1 and P2 are two points in the x-y plane, and n is a positive real number)
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Submitted by Larry
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Rating: 4.0000 (1 votes)
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Solution:
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First assume that n does not equal one.
If the coordinates of P1 are (a,b), and those of P2 are (c,d), then all points (X,Y) must satisfy:
n * sqrt((X-a)^2 + (Y-b)^2) = sqrt((X-c)^2 + (Y-d)^2)
after squaring both sides and some moderately messy algebra, this can be put in the form:
(X – ((a*n^2 – c) / (n^2 – 1)))^2 + (Y – ((b*n^2 – d) / (n^2 – 1)))^2 = (n/(n^2 – 1))^2 * (a^2+b^2+c^2+d^2 – 2ac -2bd)
which is the equation of a circle.
Of course, if n=1 then the locus of points is a line, namely the parallel bisector of the line segment between P1 P2.
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