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## Homework Statement

(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

## Homework Equations

|z|

^{2}=z.z'

## 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?