Let O be the origin and A be the point z 1 = 1 + 2 i . If B is the point z 2 , Re z 2 < 0 , such that O A B is a right angled isosceles triangle with O B as hypotenuse, then which of the following is NOT true?

Step-by-step solution

Given O be the origin and A be the point z 1 = 1 + 2 i , B is the point z 2 , Re z 2 < 0 , Also O A B is a right angled isosceles triangle with O B as hypotenuse, Plotting the diagram we get, Now by using concept of rotation we get, z 2 - 0 1 + 2 i - 0 = O B O A e i π 4 ⇒ z 2 1 + 2 i = 2 e i π 4 O r   z 2 = 1 + 2 i 1 + i = - 1 + 3 i Now finding arg z 2 = π - tan - 1 3 1 = π - tan - 1 3 Now z 2 = 1 2 + 3 2 = 10 Now solving z 1 - 2 z 2 = 1 + 2 i + 2 - 6 i = 3 - 4 i So, arg z 1 - 2 z