Let `a , b ,xa n dy` be real numbers such that `a-b=1a n dy!=0.` If the complex number `z=x+i y` satisfies `I m((a z+b)/(z+1))=y` , then which of the following is (are) possible value9s) of x?| `-1-sqrt(1-y^2)` (b) `1+sqrt(1+y^2)` `-1+sqrt(1-y^2)` (d) `-1-sqrt(1+y^2)`

Options

A. `-1-sqrt(1-y^(2))`
B. `1+sqrt(1+y^(2))`
C. `1-sqrt(1+y^(2))`
D. `-1+sqrt(1-y^(2))`

Answer

A, D

Explanation

We have,
`"Im"((az+b)/(z+1))=y`
`rArr "Im"((ax+b)+iay)/((x+1)+iy))=y`
`rArr "Im"{((ax+b)(x+1)+ay^(2))/((x+1)^(2)+y^(2))+i{(-y(ax+b)+ay(x+1))/((x+1)^(2)+y^(2))}}=y`
`rArr ((a-b)y)/((x+1)^(2)+y^(2))=y rArr (x+1)^(2)+y^(2)=1 rArr x=-1 +- sqrt(1-y^(2))`