Let z 1 and z 2 be any two non-zero complex numbers such that 3 z 1 = 4 z 2 . If z = 3 z 1 2 z 2 + 2 z 2 3 z 1 then maximum value of z is Note: In actual paper value of z was asked. Hence, none of the options given were correct. So we have modified the question as well as options.

Step-by-step solution

Let a r g 3 z 1 2 z 2 = θ , and we know that, if the a r g z = α , then a r g 1 z = - α . ∴   a r g 2 z 2 3 z 1 = - θ Also, we know that a complex number w can be expressed as w = w cos α + i sin α , where α is the argument of the complex number. ∴   z = 3 2 z 1 z 2 cos θ + i sin θ + 2 3 z 2 z 1 cos θ - i sin θ Given, 3 z 1 = 4 z 2 ,     ⇒ z 1 z 2 = 4 3 , ⇒ z = 3 2 × 4 3 cos θ + i sin θ + 2 3 × 3 4