Let complex numbers α and 1 α ¯ lie on circles x - x 0 2 + y - y 0 2 = r 2 , and x - x 0 2 + y - y 0 2 = 4 r 2 , respectively. If z 0 = x 0 + i ⁡ y 0 satisfies the equation 2 z 0 2 = r ⁡ 2 + 2 , then α =

Step-by-step solution

z - z 0 = r z - z 0 = 2 r α - z 0 = r 1 α ¯ - z 0 = 2 r α α ¯ = α 2 α α 2 - z 0 = 2 r α - z 0 α ¯ - z ¯ 0 = r 2 ⇒ α 2 - z 0 α ¯ - α z ¯ 0 + z 0 2 = r 2 α α 2 - z 0 α ¯ α 2 - z ¯ 0 = 4 r 2 ⇒ α 2 α 4 - z ¯ 0 α α 2 - z ¯ 0 α α 2 + z 0 2 = 4 r 2 1 - z 0 α ¯ - z ¯ 0 α + z 0 2 α 2 = 4 r 2 α 2 ⇒ α 2 - 1 + z 0 2 1