If z is a complex number such that z ≤ 1 , then the minimum value of z + 1 2 3 + 4 i is:

Step-by-step solution

Given: z + 3 + 4 i 2 ; z ≤ 1 Let, z = x + i y And z = 1 gives x 2 + y 2 = 1 , which represents a circle having centre at 0 , 0 and radius as 1 unit. Now, z + 3 + 4 i 2 gives the distance of z from - 3 2 , - 2 So, we need to find the minimum distance of A P . ⇒ A P = O P - O A ⇒ A P = - 3 2 - 0 2 + 0 + 2 2 - 1 ⇒ A P = 9 4 + 4 - 1 ⇒ A P = 5 2 - 1 ⇒ A P = 3 2