JEE MAIN - Mathematics (2025 - 7th April Morning Shift - No. 13)
Among the statements
(S1) : The set $\left\{z \in \mathbb{C}-\{-i\}:|z|=1\right.$ and $\frac{z-i}{z+i}$ is purely real $\}$ contains exactly two elements, and
(S2) : The set $\left\{z \in \mathbb{C}-\{-1\}:|z|=1\right.$ and $\frac{z-1}{z+1}$ is purely imaginary $\}$ contains infinitely many elements.
Explanation
$$\begin{aligned} & \frac{z-i}{z+i}=\frac{\bar{z}+i}{\bar{z}-i} \\ & =z \bar{z}-i \bar{z}-i z-1=z \bar{z}+z i+i \bar{z}-1 \\ & =z+\bar{z}=0 \\ & =2 x=0 \\ & =x=0 \quad \text { (y-axis) } \end{aligned}$$
$$\begin{aligned} & |z|=1 \\ & \therefore \quad z=i \quad(z \neq-i \text { is given }) \end{aligned}$$
Statement 1 is incorrect
$$\begin{aligned} & \frac{z-i}{z+i}+\frac{\bar{z}-1}{\bar{z}+1}=0 \\ & =z \bar{z}-\bar{z}+z-1+z \bar{z}-z+\bar{z}-1=0 \\ & =z \bar{z}=1 \\ & =|z|=1 \end{aligned}$$
Statement 2 is correct
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