Let complex numbers `alpha and 1/alpha` lies on circle `(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+iy_0` satisfies the equation `2|z_0|^2=r^2+2` then `|alpha|` is equal to (a) `1/sqrt2` (b) `1/2` (c) `1/sqrt7` (d) `1/3`

Let complex numbers `alpha and 1/alpha` lies on circle `(x-x_0)^2(y-y_

1.	Let complex numbers `alpha and 1/alpha` lies on circle `(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+iy_0` satisfies the equation `2\|z_0\|^2=r^2+2` then `\|alpha\|` is equal to (a) `1/sqrt2` (b) `1/2` (c) `1/sqrt7` (d) `1/3`
Answer» `\|alpha-Z_0\|=r-(1)` `\|1/alpha-Z_0\|=2r` `z\|z_0\|^2=r^2+2-(3)` `\|alpha/\|alpha\|^2-Z_0\|=2r-(2)` `alphaoverlinealpha=\|alpha\|^2` `1/overlinealpha=alpha/\|alpha\|^2` squaring equation 1 `\|alpha\|^2+\|Z_0\|^2-(alphaoverlineZ_0+overlinealphaZ_0)=r^2` `\|alpha\|^2+r^2/2+1-(alphaoverlineZ_0+overlinealphaZ_0)=r^2-(4)` `\|alpha\|^2/\|alpha\|^4+\|Z_0\|^2-((alphaoverlineZ_0+overlinealphaZ_0)/\|alpha\|^2)=4r^2` `1/\|alpha\|^2(1-(alphaoverlineZ_0+overlinealphaZ_0))=4r^2-\|Z_0\|^2` `=4r^2-((r^2+2)/2)` `=(7r^2)/2-1` `(1-(alphaoverlineZ_0+overlinealphaZ_o))=\|alpha\|^2[(7r^2)/2-1]` From equation 4 `\|alpha\|^2+r^2/2+1-(1+\|alpha\|^2(1-(7r^2)/2))=r^2` `r^2/2+\|alpha\|^2(7r^2)/2=r^2` `\|alpha\|^2*(7r^2/2)=r^2/2` `\|alpha\|^2=1/7` `\|alpha\|=1/sqrt7` Option C is correct.