The set of all α ∈ R , for which w = 1 + 1 - 8 α z 1 - z is a purely imaginary number, for all z ∈ C satisfying z = 1 and R e z ≠ 1 , is :

Question

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

MARKS App · Accepted Answer

For w to be purely imaginary, 1 + 1 - 8 α z 1 - z + 1 + 1 - 8 α z ¯ 1 - z ¯ = 0 ⇒ 1 - z ¯ + 1 - 8 α z - 1 - 8 α + 1 - z + 1 - 8 α z ¯ - 1 - 8 α = 0 ⇒ 2 - z + z ¯ + 1 - 8 α z + z ¯ - 2 + 16 α = 0 ⇒ 8 α z + z ¯ = 16 α ⇒ z + z ¯ = 2 or α = 0 For all z ∈ C , we have α = 0 .

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