If z = 1 2 - 2 i , is such that | z + 1 | = α z + β ( 1 + i ) , i = - 1 and α , β ∈ R , then α + β is equal to

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Correct answer: B. 3

Given: z + 1 = α z + β 1 + i , z = 1 2 - 2 i ⇒ 1 2 - 2 i + 1 = α 1 2 - 2 i + β 1 + i ⇒ 3 2 - 2 i = α 2 - 2 α i + β + β i ⇒ 9 4 + 4 = α 2 + β + i - 2 α + β Now, on comparing real and imaginary part we get, ⇒ 5 2 = α 2 + β + i - 2 α + β ⇒ 2 α = β , 5 2 = α 2 + β ⇒ 2 α = β , 5 2 = 5 α 2 ⇒ α = 1 , β = 2 ⇒ α + β = 3

If z = 1 2 - 2 i , is such that | z + 1 | = α z + β ( 1 + i ) , i = - 1 and α , β ∈ R , then α + β is equal to

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