A complex number z is rotated in anticlockwise direction by an angle `alpha` and we get `z'` and if the same complex number z is rotated by an angle `alpha` in clockwise direction and we get `z''`then

Options

A. `z',z'.z''` are in G.P
B. z',z',z'' are H.P
C. `z'+z'' = 2z cos alpha `
D. `z'^(2)+z''^(2) = 2z^(2) cos 2alpha`

Answer

A, C, D

Explanation

`z'=ze^(ialpha)" "(1)`
`z'' = ze^(-ialpha)" "(2)`
`therefore z'z''= z^(2)`
`rArr z',z,z'' ` are in G.P
Also , `((z)/(z))^(2) + ((z'')/(z)) = 2cos alpha `
`z'^(2) + z''^(2) = 2z^(2) cos 2 alpha`