If the least and the largest real values of α , for which the equation z + α z - 1 + 2 i = 0 z ∈ C and i = - 1 has a solution, are p and q respectively; then 4 p 2 + q 2 is equal to_______.

Step-by-step solution

Put z = x + i y x + i y + α x + i y - 1 + 2 i = 0 ⇒    x + α x - 1 2 + y 2 + i y + 2 = 0 + 0 i ⇒   y + 2 = 0 and x + α x - 1 2 + y 2 = 0 ⇒   y = - 2 and α 2 = x 2 x 2 - 2 x + 5 Now x 2 x 2 - 2 x + 5 ∈ 0 , 5 4 ∴   α 2 ∈ 0 , 5 4 ⇒ α ∈ - 5 2 , 5 2 ∴   p = - 5 2 ; q = 5 2 ⇒   4 p 2 + q 2 = 4 5 4 + 5 4 = 10