If the locus of the complex number z given by ar g z + i - a r g z - i = 2 π 3 is an arc of a circle, then the length of the arc is

Step-by-step solution

Given, arg z + i z - i = 2 π 3 Let, z = x + i y ⇒ x + i y + 1 x + i y - 1 = x 2 + y 2 - 1 + 2 i x x 2 + y - 1 2 ⇒ tan - 1 2 x x 2 + y 2 - 1 = 2 π 3 ⇒ x 2 + y 2 + 2 3 x - 1 = 0 Hence, the given locus is a circle with centre - 1 3 , 0 and radius 2 3 units ⇒ Length of the arc of the circle is 2 π 3 × 2 3 = 4 π 3 3 units