The set of all α ∈ R , for which w = 1 + 1 - 8 α z 1 - z is a purely imaginary number, for…

The set of all α∈R, for which w=1+1-8αz1-z is a purely imaginary number, for all z∈C satisfying z=1 and Rez≠1, is :
  1. 0
  2. 0,14,-14
  3. equal to R
  4. an empty set

Solution

For w to be purely imaginary, 1+1-8αz1-z+1+1-8αz¯1-z¯=0

⇒1-z¯+1-8αz-1-8α+1-z+1-8αz¯-1-8α=0

⇒2-z+z¯+1-8αz+z¯-2+16α=0

⇒8αz+z¯=16α

⇒z+z¯=2 or α=0

For all z∈C, we have α=0.

Asked in: JEE Main 2018 (15 Apr)

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