Question

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

• A. -1
• B. 2i
• C. -2i

Answer 1

The correct option is B 2i

Squaring the given relation, we have
$\sum \frac{{\alpha }^2}{a^2}+2(\frac{\alpha \beta }{ab}+\frac{\beta \gamma }{bc}+\frac{\gamma \alpha }{ca})=1+i^2+2i$
$\Rightarrow \sum \frac{{\alpha }^2}{a^2}=2i$ as $\sum \frac{a}{\alpha }=0$ given.