Let $\alpha, \beta, \gamma$ and $a, b, c$ be complex numbers satisfying $$ \frac{\alpha}{a}+\frac{\beta}{b}+\frac{\gamma}{c}=1+i \quad\text{and}\quad \frac{a}{\alpha}+\frac{b}{\beta}+\frac{c}{\gamma}=0. $$ Then the value of \[ \frac{\alpha^2}{a^2}+\frac{\beta^2}{b^2}+\frac{\gamma^2}{c^

Question

Let $\alpha, \beta, \gamma$ and $a, b, c$ be complex numbers satisfying $$ \frac{\alpha}{a}+\frac{\beta}{b}+\frac{\gamma}{c}=1+i \quad\text{and}\quad \frac{a}{\alpha}+\frac{b}{\beta}+\frac{c}{\gamma}=0. $$ Then the value of $$ \frac{\alpha^2}{a^2}+\frac{\beta^2}{b^2}+\frac{\gamma^2}{c^2} $$ is

Accepted Answer

0. The equations provided imply a relationship between the ratios of the complex numbers. By manipulating these ratios, particularly using the identity involving their reciprocals, we can derive that the sum of the squares of the ratios equals zero.

Let \(\alpha, \beta, \gamma\) and \(a, b, c\) be complex numbers satisfying \[ \frac{\alpha}{a}+\frac{\beta}{b}+\frac{\gamma}{c}=1+i \quad\text{and}\quad \frac{a}{\alpha}+\frac{b}{\beta}+\frac{c}{\gamma}=0. \] Then the value of \[ \frac{\alpha^2}{a^2}+\frac{\beta^2}{b^2}+\frac{\gamma^2}{c^2} \] is

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