Suppose z ∈ C has argument θ such that 0 < θ < π 2 and satisfy the equation | z - 3 i | = 3 , then what is the value of cot θ - 6 z ?

Step-by-step solution

Given, z - 3 i = 3 which represents a circle with radius as 3 and centre as ( 0 , 3 ) Now, Z is lying anywhere on z - 3 i = 3 From the figure below, we see that ∠ O A B = 90 ° (Angle subtended in a semi-circle) So, O A = Z O B = 6 O A = O B   sin θ   e i θ ⇒ Z = 6   sin θ   e i θ ⇒ 6 Z = 1 sin θ   ( e i θ )   ⇒ 6 Z = 1 sin θ   ( cos θ   + i sin θ )   By rationalising, we get ⇒ 6 Z = cos &#952