anonymous
  • anonymous
A complex question
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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schrodinger
  • schrodinger
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anonymous
  • anonymous
\[\left| z-1 \right|+\left| z+1 \right|=4\] 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.
anonymous
  • anonymous
(Where z=a+ib)
anonymous
  • anonymous
(And by 'graphically' I mean plotting the possible values of z on the complex plane)

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More answers

anonymous
  • anonymous
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?
anonymous
  • anonymous
OK
anonymous
  • anonymous
| a + bi - 1| + |a + bi + 1| = 4
anonymous
  • anonymous
| (a -1) + (b)i | + | (a+1) + (b)i | = 4
anonymous
  • anonymous
Now let me find the modulus which is given by sqrt( re(z)^2 + Im(z)^2)
anonymous
  • anonymous
\[\sqrt{(a +1)^2 + b^2} + \sqrt{(a-1)^2 + b^2} = 4\]
anonymous
  • anonymous
Then you should square both sides
anonymous
  • anonymous
Actually that will be hard
anonymous
  • anonymous
So move one of the square roots on the left then square both sides
anonymous
  • anonymous
\[(\sqrt{(a+1)^2 + b^2} )^2= (4 - \sqrt{(a-1)^2 + b^2})^2\]
anonymous
  • anonymous
Which will give you this
anonymous
  • anonymous
There's still a horrible square root sign there, though.
anonymous
  • anonymous
\[(a+1)^2 + b^2= 16 - 8\sqrt{(a-1)^2 + b^2} + (a-1)^2 + b^2\]
anonymous
  • anonymous
I will get rid of it soon
anonymous
  • anonymous
Ok now i am going to cancel some stuff and simplify this equation
anonymous
  • anonymous
\[a^2 + 2a + 1 + b^2 = 16 - 8\sqrt{(a-1)^2 + b^2} + a^2 - 2a + 1 + b^2\]
anonymous
  • anonymous
Now i can simplify further
anonymous
  • anonymous
\[4a -16 = -8\sqrt{(a-1)^2+b^2}\]
anonymous
  • anonymous
4a-16=-8sqrt(a^2-2a+1+b^2)
anonymous
  • anonymous
Now I am going to square both sides again
anonymous
  • anonymous
\[(4a -16)^2 = 64((a-1)^2 + b^2)\]
anonymous
  • anonymous
Finally we have no more square roots
anonymous
  • anonymous
Now i am going to expand
anonymous
  • anonymous
\[16a^2 -128a + 256 = 64 a^2 -128a + 64 + 64b^2)\]
anonymous
  • anonymous
it ends up as 192=48a^2+64b^2 I think
anonymous
  • anonymous
Now simlify...
anonymous
  • anonymous
SSo you get
anonymous
  • anonymous
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!
anonymous
  • anonymous
Final step: complete the square on a term
anonymous
  • anonymous
oops
anonymous
  • anonymous
Sorry yeah that was the answer 192 = 48a^2 +64b^2
anonymous
  • anonymous
And thats an eclipse.
anonymous
  • anonymous
No problem. =)

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