Let the complex numbers α and 1 α ¯ lie on the circles z - z 0 2 = 4 and z - z 0 2 = 16 respectively, where z 0 = 1 + i . Then, the value of 100 | α | 2 is__________.

Step-by-step solution

Given: z - z 0 2 = 4 ⇒ z - z 0 z - z 0 = 4 ⇒ z z - z 0 z - z 0 z + z 0 2 = 4 ⇒ z 2 - z 0 z - z 0 z = 2 as z 0 2 = 2 It is given that, α lies on z - z 0 2 = 4 . ⇒ α 2 - α z 0 - z 0 α - 2 = 0 . . . i We know that, α · α = α 2 = α 2 z - z 0 2 = 16 ⇒ 1 α ¯ - z 0 1 α - z ¯ 0 = 16 ⇒ 1 - α ¯ z 0 1 - α z ¯ 0 = 16 | α | 2 ⇒ 1 - α ¯ z 0 - α z ¯ 0 + | α | 2 z 0 2 = 16 | α | 2 ⇒ 1 - α ¯ z 0 - α z ¯ 0 = 14 | α | 2 . . . ii Using, i and ii ⇒ 5 | α | 2 = 1 ⇒ 100 | α | 2 = 20