The set of all R , for which w= 1+(1-8 ) z 1-z is a purely imaginary number, for all z C satisfying |z|=1 and Re z 1 , is

Step-by-step solution

aligned & |z|=1 & Re z 1 & Suppose z=x+i y x^2+y^2=1 ( i ) & Now, w= 1+(1-8 ) z 1-z & w= 1+(1-8 )(x+i y) 1-(x+i y) & w= 1+(1-8 )(x+i y))((1- x )+ iy ) 1-(x+i y))((1- x )+ iy ) & w= [ (1+x(1-8 )(1-x)-(1-8) y^2 ] . (1-x)^2+y^2 &+i [(1+x(1-8 )) y-(1-8 ) y(1-x)] (1-x)^2+y^2 aligned If, w is purely imaginary. So, aligned & Re w= [(1+x(1-8 ))(1- )-(1-8 ) y^2 ] (1-x)^2+y^2 &=0 & (1-x)+x(1-8 )(1-x)=(1-8) y^2 & (1-x)+x(1-8 )-x^2(1-8 )=(1-8 x) y^2 & (1-x)+x(1-8 )=1-8 & [ From (i), x^2+y^2=1 ] aligned 1-8 =1 =0 0