Let C be the circle in the complex plane with centre z 0 = 1 2 1 + 3 i and radius r = 1 . Let z 1 = 1 + i and the complex number z 2 be outside circle C such that z 1 - z 0 z 2 - z 0 = 1 . If z 0 , z 1 and z 2 are collinear, then the smaller value of z 2 2 is equal to

Step-by-step solution

Given, z 0 = 1 + 3 i 2 ,   z 1 = 1 + i So, z 1 - z 0 = 1 - 1 2 2 + 1 - 3 2 2 = 1 4 + 1 4 = 1 2 And z 2 be outside circle, z 1 - z 0 z 2 - z 0 = 1 ⇒ 1 2 z 2 - z 0 = 1 ⇒ z 2 - z 0 = 2 Now given, z 0 ,   z 1   &   z 2 are collinear, So, by concept of rotation we get, z 2 - z 0 z 1 - z 0 = z 2 - z 0 z 1 - z 0 e i 0 ⇒ z 2 - z 0 z 1 - z 0 = z 2 - z 0 z 1 - z 0 ± 1 ⇒ z 2 - z 0 z 1 - z 0 = ± 2 ⇒ z 2 = z 0 ± 2 z 1 - z 0 So, z 2 = 2 z 1 - z 0 = 3 2 + 1 2 i