For a complex number Z , if the argument of Z - a Z ¯ - b is π 4 or - 3 π 4 (where a , b are two real numbers), then the value of a b such that the locus of Z represents a circle with centre 3 2 + i 2 is (where, i 2 = - 1 )

Step-by-step solution

Let, Z = x + i y , then Z - a Z ¯ - b = Z Z ¯ - a Z ¯ - b Z + a b = x 2 + y 2 - a x - i y - b x + i y + a b = x 2 + y 2 - a + b x + i a - b y + a b If the argument of the above complex number is π 4 or - 3 π 4 , then x 2 + y 2 - a + b x + a b = a - b y ⇒ Centre of the circle is a + b 2 , a - b 2 ⇒ a + b 2 = 3 2 & a - b 2 = 1 2 ⇒ a = 2 , b = 1 ⇒ a b = 2