Question

Find z so that
Iz+i|=|z| and
arg (z+i/z)=pi/4


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Answer 1

Answer:

z+iz−i=(z+i)(z¯¯¯+i)(z−i)(z¯¯¯+i)=zz¯¯¯+i(z+z¯¯¯)−1zz¯¯¯+i(z−z¯¯¯)+1

If z=x+iy then z¯¯¯=x−iy,z+z¯¯¯=2x,z−z¯¯¯=2iy,zz¯¯¯=x2+y2 and the fraction simplifies to

z+iz−i=x2+y2−1+i2xx2+y2+1−2y

As the denominator is positive real this will not change the argument of the fraction. Now if arg(w)=π/4 then w=r(1+i) for some r>0. So the real and imaginary parts of the numerator must be equal, and both positive, that is x2+y2−1=2x. Rearrange x2−2x+1+y2=2 and (x−1)2+y2=2 a circle center (1,0) with radius 2–√.