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` and `z = x+iy`
`therefore Im((a(x+iy)+b)/(x+iy+1))= y`
`rArr IM(((ax+b+iay)(x+1-iy))/((x +1)^(2) +y^(2)))`
`rArr -y(ax + b)+ay(x+1)=y((x+1)^(2)+y^(2))`
`because y ne 0 and a-b=1`
`therefore (x+1)^(2) +y^(2) = 1`
`rArr x = - 1 pm sqrt(1-y^(2))`