Let z be the complex number satisfying |z-5| 3 and having maximum positive principal argument. Then 34 | 5 z-12 5 i z+16 |² is equal to :

Step-by-step solution

The region |z - 5| 3 is a disk centered at (5, 0) with radius 3. Maximum positive principal argument from origin occurs at the tangent point from origin to the circle. = 3 5 , = 4 5 . The tangent point is the foot of perpendicular from (5,0) to the line 3x - 4y = 0 , giving z = 16 5 + 12 5 i . 5z - 12 = 4 + 12i |5z - 12|^2 = 160 5iz + 16 = 4 + 16i |5iz + 16|^2 = 272 34 | 5z-12 5iz+16 |^2 = 34 160 272 = 34 10 17 = 20 .