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NDA 1 2024 Maths, question 9

Topic: Algebra

Q9

Let z1z_{1} and z2z_{2} be two complex numbers such that ∣z1+z2z1−z2∣=1\left| \frac{z_{1}+z_{2}}{z_{1}-z_{2}} \right|=1, then what is Re(z1z2)+1Re\left( \frac{z_{1}}{z_{2}} \right)+1 equal to ?

  1. (a) −1-1
  2. (b) 00
  3. (c) 11Correct answer
  4. (d) 55

Solution.

Answer: (c) 11

  1. The condition gives |z₁+z₂|²=|z₁−z₂|², hence 4Re(z₁ conjugate(z₂))=0.
  2. For the requested ratio, z₂≠0, so Re(z₁/z₂)=Re(z₁ conjugate(z₂))/|z₂|²=0.
  3. Adding 1 gives 1.