Let \[a,b,c\] be real numbers \[a\ne 0\]. If \[\alpha \]is a root \[{{a}^{2}}{{x}^{2}}+bx+c=0\], \[\beta \] is a root of \[{{a}^{2}}{{x}^{2}}-bx-c=0\] and \[0<\alpha <\beta \], then the equation \[{{a}^{2}}{{x}^{2}}+2bx+2c=0\]has a root \[\gamma \]that always satisfies [IIT 1989]

• A. \[\gamma =\frac{\alpha +\beta }{2}\]
• B. \[\gamma =\alpha +\frac{\beta }{2}\]
• C. \[\gamma =\alpha \]
• D. \[\alpha <\gamma <\beta \]