I have to find all z∈C for which BOTH of the following is true:
1) |z|=1
2) |z4+1|=1
I understand that the 1) is a unit circle, but I can't find out what would be the 2).
Calculating the power of (x+yi) seems wrong because I get a function with square root of polynomial of the 8th degree.
I've even tried to equate the two, since their modulo is the same, like this:
|z|=|z4+1|
and then separating two possibilities in which the equation holds, but I get only
z4±z+1=0
which I don't know how to calculate.
I guess there is an elegant solution which I am unable to see...
Any help?
Answer
It seems helpful to square both sides. From the second equation, you get 1=(z4+1)(¯z4+1)=(z¯z)4+z4+¯z4+1=z4+¯z4+2. That is, z4+¯z4=−1.
The first equation lets you write z=eiθ, so you have −1=e4iθ+e−4iθ=2cos(4θ); in other words, cos(4θ)=−12. The solutions of this are well-known.
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