Among the statements (S1) : The set \(\left{\mathrm{z}\in \mathrm{ℂ}-{-\mathrm{i}}:|\mathrm{z}|=1\right.\) and \(\frac{\…

\(\frac{z+i}{z−i}=\frac{x+i(y+1)}{x+i(y−1)}\times \frac{x−i(y−1)}{x−i(y−1)}\\ =\text{purely real}\)

\(=\frac{({x}^{2}+{y}^{2}−1)+i(x(y+1)−x(y−1))}{{x}^{2}+(y−1{)}^{2}}\)

Imaginary part is zero.

\(x(y+1)=x(y−1)\text{ implies }x=0\text{ and }y\text{ is}\\ \text{ infinitely many values}\)

Therefore statement 1 is wrong.

\(\frac{z−1}{z+1}=\frac{x+iy−1}{x+iy+1}\\ =\frac{(x−1)+iy}{(x+1)+iy}\times \frac{(x+1)−iy}{(x+1)−iy}\)

\(=\frac{(({x}^{2}−1){y}^{2}+i(y(x+1)−y(x−1)))}{{(x+1)}^{2}+{y}^{2}}\)

Purely imaginary implies real part is zero.

\(∴{x}^{2}+{y}^{2}=1\)

Lies on infinitely many points

Therefore statement "2" is true.