Concept of complex number

Question

Let complex numbers !! alpha !! and !!1/ baralpha !!
lie on circles !! (x - x_0 )^2!! !!+!! !! (y - y_0 )^2 !! !!=!! !!r^2!! and !!(x - x_0)^2+(y- y_0)^2=4r^2!!,respectively.

If !! z_0 = x_0 + i y_0 !! satisfies the equation !!2|z_0|^2 = r^2 + 2!!, then !!| alpha | = ?!!

(a) !!1/3!!

(b) !!1/sqrt2!!

(c) !!1/2!!

(d) !!1/sqrt7!!

Solution

!!|alpha-x_0-iy_0|=r!!

!!|alpha-Z_0|=r!!

!!|1/baralpha-x_0-iy_0|=2r!!

!!|1/baralpha-Z_0|=2r!!

Given that, !!2|Z_0|^2=r^2+2!!

We know that !!ZbarZ= |Z|^2!!

!!=>(alpha-Z_0)(baralpha-barZ_0)=r^2!!

!!=>alphabarZ_0+baralphaZ_0=|alpha|^2+1-r^2/2!!

!!=>(1/baralpha-Z_0)(1/alpha-barZ_0)=4r^2!!

!!=>(1-baralphaZ_0)(1-alphabarZ_0)=4baralphaalphar^2!!

!!=>-alphabarZ_0-baralphaZ_0=4|alpha|^2r^2-1-|alpha|^2(r^2/2+1)!!

!!=>-alphabarZ_0-baralphaZ_0=7|alpha|^2r^2/2-1-|alpha|^2!!

!!-|alpha|^2-1+r^2/2=7|alpha|^2r^2/2-1-|alpha|^2!!

!!|alpha|=1/sqrt(7)!!

Therefore, option (d) is correct.