If |z+i|=|z| and arg(z+i/z)= π/4, then find z.
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First multiply top and bottom by the complex conjugate of the denominator
z+iz−i=(z+i)(z¯¯¯+i)(z−i)(z¯¯¯+i)=zz¯¯¯+i(z+z¯¯¯)−1zz¯¯¯+i(z−z¯¯¯)+1z+iz−i=(z+i)(z¯+i)(z−i)(z¯+i)=zz¯+i(z+z¯)−1zz¯+i(z−z¯)+1
If z=x+iyz=x+iy then z¯¯¯=x−iy,z+z¯¯¯=2x,z−z¯¯¯=2iy,zz¯¯¯=x2+y2z¯=x−iy,z+z¯=2x,z−z¯=2iy,zz¯=x2+y2and the fraction simplifies to
z+iz−i=x2+y2−1+i2xx2+y2+1−2yz+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/4arg(w)=π/4 then w=r(1+i)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=2xx2+y2−1=2x. Rearrange x2−2x+1+y2=2x2−2x+1+y2=2 and (x−1)2+y2=2(x−1)2+y2=2 a circle center (1,0) with radius 2–√2.
There might be a neater way of showing this, the formula is related to Möbius transformation. These transformations have the general form az+bcz+daz+bcz+d and preserve angles, map every straight line to a line or circle, and map every circle to a line or circle. The inverse transformation will also map straight lines, like the line arg(w)=π/4arg(w)=π/4, to circles.
z+iz−i=(z+i)(z¯¯¯+i)(z−i)(z¯¯¯+i)=zz¯¯¯+i(z+z¯¯¯)−1zz¯¯¯+i(z−z¯¯¯)+1z+iz−i=(z+i)(z¯+i)(z−i)(z¯+i)=zz¯+i(z+z¯)−1zz¯+i(z−z¯)+1
If z=x+iyz=x+iy then z¯¯¯=x−iy,z+z¯¯¯=2x,z−z¯¯¯=2iy,zz¯¯¯=x2+y2z¯=x−iy,z+z¯=2x,z−z¯=2iy,zz¯=x2+y2and the fraction simplifies to
z+iz−i=x2+y2−1+i2xx2+y2+1−2yz+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/4arg(w)=π/4 then w=r(1+i)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=2xx2+y2−1=2x. Rearrange x2−2x+1+y2=2x2−2x+1+y2=2 and (x−1)2+y2=2(x−1)2+y2=2 a circle center (1,0) with radius 2–√2.
There might be a neater way of showing this, the formula is related to Möbius transformation. These transformations have the general form az+bcz+daz+bcz+d and preserve angles, map every straight line to a line or circle, and map every circle to a line or circle. The inverse transformation will also map straight lines, like the line arg(w)=π/4arg(w)=π/4, to circles.
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