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henpen

  • 2 years ago

A complex question

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  1. henpen
    • 2 years ago
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    \[\left| z-1 \right|+\left| z+1 \right|=4\] Which turns into the messy \[\sqrt{(a-1)^2+b^2}+\sqrt{(a+1)^2+b^2}=4\] Which I have no idea about how to solve graphically.

  2. henpen
    • 2 years ago
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    (Where z=a+ib)

  3. henpen
    • 2 years ago
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    (And by 'graphically' I mean plotting the possible values of z on the complex plane)

  4. henpen
    • 2 years ago
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    http://www.wolframalpha.com/input/?i=sqrt%28%28x%2B1%29%5E2%2By%5E2%29%2Bsqrt%28%28x-1%29%5E2%2By%5E2%29%3D4 Can anyone explain why the graph is an ellipse?

  5. baddinlol
    • 2 years ago
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    OK

  6. baddinlol
    • 2 years ago
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    | a + bi - 1| + |a + bi + 1| = 4

  7. baddinlol
    • 2 years ago
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    | (a -1) + (b)i | + | (a+1) + (b)i | = 4

  8. baddinlol
    • 2 years ago
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    Now let me find the modulus which is given by sqrt( re(z)^2 + Im(z)^2)

  9. baddinlol
    • 2 years ago
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    \[\sqrt{(a +1)^2 + b^2} + \sqrt{(a-1)^2 + b^2} = 4\]

  10. baddinlol
    • 2 years ago
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    Then you should square both sides

  11. baddinlol
    • 2 years ago
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    Actually that will be hard

  12. baddinlol
    • 2 years ago
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    So move one of the square roots on the left then square both sides

  13. baddinlol
    • 2 years ago
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    \[(\sqrt{(a+1)^2 + b^2} )^2= (4 - \sqrt{(a-1)^2 + b^2})^2\]

  14. baddinlol
    • 2 years ago
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    Which will give you this

  15. henpen
    • 2 years ago
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    There's still a horrible square root sign there, though.

  16. baddinlol
    • 2 years ago
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    \[(a+1)^2 + b^2= 16 - 8\sqrt{(a-1)^2 + b^2} + (a-1)^2 + b^2\]

  17. baddinlol
    • 2 years ago
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    I will get rid of it soon

  18. baddinlol
    • 2 years ago
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    Ok now i am going to cancel some stuff and simplify this equation

  19. baddinlol
    • 2 years ago
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    \[a^2 + 2a + 1 + b^2 = 16 - 8\sqrt{(a-1)^2 + b^2} + a^2 - 2a + 1 + b^2\]

  20. baddinlol
    • 2 years ago
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    Now i can simplify further

  21. baddinlol
    • 2 years ago
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    \[4a -16 = -8\sqrt{(a-1)^2+b^2}\]

  22. henpen
    • 2 years ago
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    4a-16=-8sqrt(a^2-2a+1+b^2)

  23. baddinlol
    • 2 years ago
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    Now I am going to square both sides again

  24. baddinlol
    • 2 years ago
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    \[(4a -16)^2 = 64((a-1)^2 + b^2)\]

  25. baddinlol
    • 2 years ago
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    Finally we have no more square roots

  26. baddinlol
    • 2 years ago
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    Now i am going to expand

  27. baddinlol
    • 2 years ago
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    \[16a^2 -128a + 256 = 64 a^2 -128a + 64 + 64b^2)\]

  28. henpen
    • 2 years ago
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    it ends up as 192=48a^2+64b^2 I think

  29. baddinlol
    • 2 years ago
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    Now simlify...

  30. baddinlol
    • 2 years ago
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    SSo you get

  31. henpen
    • 2 years ago
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    That actually agrees with the wolfram link I posted earlier. Thanks very much- I had little idea how to deal with square roots efficiently before!

  32. baddinlol
    • 2 years ago
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    Final step: complete the square on a term

  33. baddinlol
    • 2 years ago
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    oops

  34. baddinlol
    • 2 years ago
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    Sorry yeah that was the answer 192 = 48a^2 +64b^2

  35. baddinlol
    • 2 years ago
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    And thats an eclipse.

  36. baddinlol
    • 2 years ago
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    No problem. =)

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