If for the complex numbers z satisfying | z - 2 - 2 i | ≤ 1 , the maximum value of | 3 i z + 6 | is attained at a + i b , then a + b is equal to _____ .

Step-by-step solution

Given that: | z - 2 - 2 i | ≤ 1 which represents a circle with centre at ( 2 , 2 ) and radius = 1 . Now | 3 i z + 6 | = | 3 i ( z - 2 i ) | = | 3 i | | z - 2 i | As we know that,   | z 1 z 2 | = | z 1 | | z 2 | From tha diagram we can see that, | 3 i z + 6 | = 3 | z - 2 i | is maximum at z = 3 + 2 i because a point on the circle, which has maximum distance from ( 0 , 2 ) is ( 3 , 2 ) . So, a = 3   &   b = 2 ⇒ a + b = 5