(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

(z' represent conjugate of complex number z,i is iota =sqrt(-1))

(1)find the locus of z.

|z|^{2}+4z'=0

(2)|z_{1}|=1, |z_{2}|=2, then find the value of |z_{1}-z_{2}|^{2}+|z_{1}+z_{2}|^{2}

(3)z=(k+3)+i[sqrt(3-k^{2})] for all real k. find locus of z

2. Relevant equations

|z|^{2}=z.z'

3. The attempt at a solution

(1) |z|^{2}+4z'= |z|^{2}+4z'*z/z

|z|^{2}{1+4/z}=0

it is possible iff z= -4. so it has one solution.

(2) I take z_{1}= cos(a)+i sin(a) and z_{2}=2{cos(b)+i*sin(b)}

after putting these values in the required equation i got 10.

(3) let z=x+iy

x=k+3; y=sqrt(3-k^{2})

x-3=k; y^{2}=3-k^{2}

squaring x and adding it to y^{2},

(x-3)^{2}+y^{2}=k^{2}+3-k^{2}

that's circle.

unfortunately all of my answers are incorrect.

I want to know why?

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# Basic questions of complex number's geometry?

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