If $$\alpha ,\beta ,\gamma $$  and $$a, b, c$$  are complex numbers such that $$\frac{\alpha }{a} + \frac{\beta }{b} + \frac{\gamma }{c} = 1 + i$$     and $$\frac{a}{\alpha } + \frac{b}{\beta } + \frac{c}{\gamma } = 0,$$    then the value of $$\frac{\alpha^2 }{a^2} + \frac{\beta^2 }{b^2} + \frac{\gamma^2 }{c^2}$$   is equal to

Solution :
$$\eqalign{ & \frac{\alpha }{a} + \frac{\beta }{b} + \frac{\gamma }{c} = 1 + i{\text{ squaring}} \cr & \frac{{{\alpha ^2}}}{{{a^2}}} + \frac{{{\beta ^2}}}{{{b^2}}} + \frac{{{\gamma ^2}}}{{{c^2}}} + 2\left( {\frac{{\alpha \beta }}{{ab}} + \frac{{\beta \gamma }}{{bc}} + \frac{{\gamma \alpha }}{{ac}}} \right) = 2i \cr & {\text{or }}\frac{{{\alpha ^2}}}{{{a^2}}} + \frac{{{\beta ^2}}}{{{b^2}}} + \frac{{{\gamma ^2}}}{{{c^2}}} + \frac{{2\alpha \beta \gamma }}{{abc}}\left( {\frac{c}{\gamma } + \frac{a}{\alpha } + \frac{b}{\beta }} \right) = 2i \cr & \therefore \frac{{{\alpha ^2}}}{{{a^2}}} + \frac{{{\beta ^2}}}{{{b^2}}} + \frac{{{\gamma ^2}}}{{{c^2}}} = 2i \cr} $$