If `z_(1)`, `z_(2)` are two complex numbers such that `|(z_(1)-z_(2))/(z_(1)+z_(2))|=1` and `iz_(1)=Kz_(2)`, where ` K in R`, then the angle between `
If `z_(1)`, `z_(2)` are two complex numbers such that `|(z_(1)-z_(2))/(z_(1)+z_(2))|=1` and `iz_(1)=Kz_(2)`, where ` K in R`, then the angle between `z_(1)-z_(2)` and `z_(1)+z_(2)` is
A. `tan^(-1)((2K)/(K^(2)+1))`
B. `tan^(-1)((2K)/(1-K^(2)))`
C. `-2tan^(-1)K`
D. `2tan^(-1)K`
1 Answers
Correct Answer - D
`(d)` `(z_(1)-z_(2))/(z_(1)+z_(2))=cosalpha+isinalpha`
`implies(2z_(1))/(-2z_(2))=(cosalpha+isinalpha+1)/(cosalpha-1+isinalpha)`
`(2cos^(2)alpha//2+2isinalpha//2cosalpha//2)/(2isinalpha//2cosalpha//2-2sin^(2)alpha//2)`
`(2cosalpha//2[cosalpha//2+isinalpha//2])/(2isinalpha//2[cosalpha//2+isinalpha//2])`
`implies(z_(1))/(z_(2))=icot"(alpha)/(2)`
`implies` Given `(z_(1))/(z_(2))=(K)/(1)`
`:.tanalpha//2=-1//K`
`tanalpha=(2tanalpha//2)/(1-tan^(2)alpha//2)implies(-2//K)/(1-1//K^(2))implies(-2K)/(K^(2)-1)`
`alpha=tan^(-1)((2K)/(1-K^(2)))implies2tan^(-1)(K)`