Let `|z|=2and w=(z+1)/(z-1),where z ,w , in C` (where `C` is the set of complex numbers). Then product of least and greatest value of modulus of `w` is__________.

Answer

1

Explanation

Let `z = a+ib`
Given `|z| = 2`
`rArr a^(2) + b^(2) = 4 rArr a,b in [-2,2]`
Now `w ((a+1)+ib)/((a-1)+ib),` `rArr |w| = sqrt(((a+i)^(2) + b^(2))/((a-1)^(2) + b^(2)))=sqrt((a^(2) + b^(2)+2a+1)/(a^(2) +b^(2) -2a+1)) = sqrt((5+2a)/(5-2a))`
`|w|_("min") = sqrt((5+4)/(1)) = 3` (when a =2)
`|w|_("min") = sqrt((5-4)/(9)) = (1)/(3)` (when a = -2)
Hence, required product is 1.