JEE Main: How Many Solutions on |z| = 1? Complex Number Trick 💡
❓ Question Among the following statements: (S1) { z ∈ C ∖ { − i } : ∣ z ∣ = 1 and z − i z + i is purely real } contains exactly two elements . (S2) { z ∈ C ∖ { − 1 } : ∣ z ∣ = 1 and z − 1 z + 1 is purely imaginary } contains infinitely many elements . Determine which statement(s) is/are true . 🖼️ Question Image ✍️ Short Solution This question is based on a very powerful JEE concept : For a complex number z − a z − b Purely real ⇔ arguments are equal or differ by π Purely imaginary ⇔ arguments differ by π 2 \frac{\pi}{2} We’ll use argument geometry on the unit circle . 🔹 Key Concept (Must Know) For any non-zero complex number w w w : w w is purely real ⇔ arg ( w ) = 0 or π w w is purely imaginary ⇔ arg ( w ) = ± π 2 🔍 Statement (S1) Analysis Condition: ∣ z ∣ = 1 , z − i z + i ∈ R |z| = 1,\quad \frac{z - i}{z + i} \in \mathbb{R} Step 1 — Convert to arg...