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Topic: Absolute And Constant Puzzle (Read 3017 times) |
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K Sengupta
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Absolute And Constant Puzzle
« on: Jun 12th, 2008, 7:33am » |
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The pair (P, Q) of real numbers satisfy the following system of equations:
2|P| + |P| = Q + P2 + M, and P2 + Q2 = 1 ………(*)
where M is a real constant.
Determine all possible values of M such that the system of equations in (*) admits of precisely one solution in (P, Q).
Note: |P| denotes the absolute value of P.
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| « Last Edit: Jun 12th, 2008, 7:36am by K Sengupta » |
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pex
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Re: Absolute And Constant Puzzle
« Reply #1 on: Jun 12th, 2008, 8:33am » |
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on Jun 12th, 2008, 7:33am, K Sengupta wrote:The pair (P, Q) of real numbers satisfy the following system of equations:
2|P| + |P| = Q + P2 + M, and P2 + Q2 = 1 ………(*)
where M is a real constant.
Determine all possible values of M such that the system of equations in (*) admits of precisely one solution in (P, Q).
Note: |P| denotes the absolute value of P.
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I think only M = 0, (P, Q) = (0, 1) and M = 2, (P, Q) = (0, -1) work, because P only appears through |P| and P2, making P and -P interchangeable.
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Eigenray
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Re: Absolute And Constant Puzzle
« Reply #2 on: Jun 12th, 2008, 9:38am » |
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m=0,2 are the only values for which there are an odd number of solutions, but only m=0 has a unique solution.
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SMQ
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Re: Absolute And Constant Puzzle
« Reply #3 on: Jun 12th, 2008, 9:43am » |
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Only M = 0 has a unique solution in both P and Q: P = 0, Q = 1. As pex points out, P must be zero (and therefore Q = +/-1) or there are multiple solutions in P, but also M must be < 2 or there are multiple solutions in Q, for instance M = 2 can be solved by P = +/-1, Q = 0 as well as P = 0, Q = -1.
(The maximum value of M (around 2.557) clearly does have a unique solution in Q but it admits two solutions in P with opposite signs.)
Edit: dang, ninja'd by Eigenray. 
--SMQ
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| « Last Edit: Jun 12th, 2008, 9:44am by SMQ » |
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--SMQ
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pex
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Re: Absolute And Constant Puzzle
« Reply #4 on: Jun 12th, 2008, 9:56am » |
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Okay, maybe I should think a bit more before posting.
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