Let a circle C in complex plane pass through the points z 1 = 3 + 4 i , z 2 = 4 + 3 i and z 3 = 5 i . If z ≠ z 1 is a point on C such that the line through z and z 1 is perpendicular to the line through z 2 and z 3 , then arg z is equal to

Step-by-step solution

z 1 ≡ A ≡ 3 , 4 ,   z 2 ≡ B ≡ 4 , 3 ,   z 3 ≡ C ≡ 0 , 5 and z ≡ P ≡ x , y Slope of B C = 3 - 5 4 - 0 = - 1 2 Slope of A P = 2 Now, equation of AP   is   y - 4 = 2 x - 3 ⇒ y = 2 x - 1 Since P lies on circle x 2 + y 2 = 25 ⇒ x 2 + 2 x - 1 2 = 25 ⇒ x = - 7 5 and y = - 24 5 ⇒ arg ( z ) = tan - 1 24 7 - π