If z₁, z₂ are two distinct complex number such that |(z₁-2 z₂)/(1/2-z₁ z̄₂)|=2, then

Solution

Maths sir · 3 steps

Clear the fraction, square both moduli, and watch the cross terms cancel so the condition factorises.

1. Clear the fraction. Multiply across: |z₁ - 2z₂| = 2|1/2 - z₁z̄₂| = |1 - 2z₁z̄₂|. Now square both sides using |a-b|² = |a|² + |b|² - 2 Re(ab̄).

   |z₁|² + 4|z₂|² - 4 Re(z₁z̄₂) = 1 + 4|z₁|²|z₂|² - 4 Re(z₁z̄₂)

2. Cancel and factorise. The Re(z₁z̄₂) terms are equal on both sides, so they cancel. That leaves |z₁|² + 4|z₂|² - 1 - 4|z₁|²|z₂|² = 0, which factorises as (|z₁|² - 1)(1 - 4|z₂|²) = 0.
3. So. A product is zero when at least one factor is zero. So either |z₁| = 1 or |z₂| = 1/2: z₁ lies on a circle of radius 1, or z₂ lies on a circle of radius 1/2. That is option (c).

The idea to keep. When a modulus equation has a conjugate product like z₁z̄₂, square both sides. The real-part cross terms usually cancel and the rest factorises into a product of moduli conditions.

Answer from the NTA final answer key. Solution by PrepWiser.