System a + b + c = 4, a 2 + b 2 + c 2 = 8 . find all possible values for c .

Question

System   a  +  b  +  c  =  4,       a    2    +      b    2    +      c    2    =  8 . find all possible values for    c .

Dobermann82 · Accepted Answer

We will show that any   c in the closed interval   [  0  ,  8  /  3  ] is achievable, and nothing else is. Let   c be any real number, and suppose   a and   b are real numbers such that   (  a  ,  b  ,  c  ) satisfies our two equations. Note that in general  2  (      a    2    +      b    2    )  −  (  a  +  b      )    2    =  (  a  −  b      )    2    ≥  0.      (  ∗  )Put       a    2    +      b    2    =  8  −      c    2   and   a  +  b  =  4  −  c. Thus from   (  ∗  )  2  (  8  −      c    2    )  −  (  4  −  c      )    2    =  (  a  −  b      )    2    ≥  0.      (  ∗  ∗  )So we must have   8  c  −  3      c    2    ≥  0. This is only true for   0  ≤  c  ≤  8  /  3.Now we show that any c in the interval   [  0  ,  8  /  3  ] is achievable. We want to make   (  a  −  b      )    2    =  8  c  −  3      c    2  . If c is in   [  0  ,  8  /  3  ], then   8  c  −  3      c    2    ≥  0, so we can take the square root, and obtain  a  −  b  =  ±      8    c    −    3          c      2        .We can now solve the system   a  −  b  =  ±      8    c    −    3          c      2      ,   a  +  b  =  4 to find the values of   a and   b. We might as well give the solutions   (  a  ,  b  ,  c  ) explicitly. They are  a  =  2  ±      1    2        8    t    −          t      2        ,    b  =  2  ∓      1    2        8    t    −          t      2        ,    c  =  t  ,where   t is a parameter that ranges over the interval   0  ≤  t  ≤  8  /  3. There is something mildly ugly in the above general solution, since it breaks symmetry. That can be fixed.

grenivkah3z · Accepted Answer

All the points are given by      (          4      3        ,              4      3        ,              4      3            )    +            12        3          (    1    ,        −    1    ,        0        )      cos  ⁡  t  +      2    3          (    1    ,        1    ,        −    2        )      sin  ⁡  twhere the points with a zero and a pair of 2&#039;s occur at   t  =      π    2    ,            7      π        6    ,            11      π        6    , and the third component is given by      4    3    (  1  −  sin  ⁡  t  )which varies between   0 and       8    3    ,, the latter happening at   t  =            3      π        2    ..

System a + b + c = 4 , a 2 </msup> + b 2

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