The value of λ for which the loci arg z = π 6 and | z - 2 3 i | = λ on the argand plane touch each other is

Step-by-step solution

z = x + i y arg z = tan - 1 y x arg z = π 6 ∴   tan - 1 y x = π 6 y x = tan π 6 = 1 3 ⇒   x = 3 y Also given, | z - 2 3 i | = λ | x + i y - 2 3 i | = λ x 2 + y - 2 3 2 = λ 2 This is the equation of circle whose centre x - 3 y = 0 is ( 0 , 2 3 ) and r = λ . Now, the line touches the circle ∴   λ = 0 - 3 2 3 1 + 3 λ = 6 2 = 3 ∴   λ = 3