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The Day of the Locus (Posted on 2006-12-07) Difficulty: 3 of 5
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)

  Submitted by Larry    
Rating: 4.0000 (1 votes)
Solution: (Hide)
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.

Comments: ( You must be logged in to post comments.)
  Subject Author Date
Some ThoughtsPuzzle Thoughts K Sengupta2023-02-13 21:47:38
SolutionsolutionCharlie2006-12-07 10:48:33
SolutionSolutionBractals2006-12-07 10:30:49
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