Question

please answer this question​

Answer 1

Given that the variables are complex numbers, the value of the complex expression $\frac{\gamma}{\alpha} +\bar{\frac{\alpha}{\beta}}$ is -2

How to solve the complex expression?

The given parameters are:

$\alpha + \beta + \gamma = 0$

$\frac{1}{\alpha} + \frac{1}{\beta} + \frac{1}{\gamma} = 0$

Rewrite the first equation as:

$\alpha + \beta = - \gamma$

Take LCM in the second equation

$\frac{\alpha + \beta}{\alpha \beta} + \frac{1}{\gamma} = 0$

So, we have:

$\frac{ - \gamma}{\alpha \beta} + \frac{1}{\gamma} = 0$

Rewrite as:

$\frac{1}{\gamma} = \frac{\gamma}{\alpha \beta}$

Multiply through by $\alpha$

$\frac{\alpha}{\gamma} = \frac{\gamma}{\beta}$

Inverse both sides

$\frac{\gamma}{\alpha} = \frac{\beta}{\gamma}$

Make $\beta$ the subject in $\alpha + \beta + \gamma = 0$

$\beta =-(\alpha + \gamma)$

So, we have:

$\frac{\gamma}{\alpha} = \frac{-(\alpha + \gamma)}{\gamma}$

Expand

$\frac{\gamma}{\alpha} = -\frac{\alpha}{\gamma}- 1$

This gives

$\frac{\gamma}{\alpha} +\frac{\alpha}{\gamma} = - 1$

Make $\gamma$ the subject in $\alpha + \beta + \gamma = 0$

$\gamma = -(\beta + \alpha)$

So, we have:

$\frac{\gamma}{\alpha} +\frac{\alpha}{-(\beta + \alpha)} = - 1$

Split

$\frac{\gamma}{\alpha} +\bar{\frac{\alpha}{\beta}} + \frac{\alpha}{\alpha} = - 1$

Evaluate the quotient

$\frac{\gamma}{\alpha} +\bar{\frac{\alpha}{\beta}} + 1 = - 1$

Subtract 1 from both sides

$\frac{\gamma}{\alpha} +\bar{\frac{\alpha}{\beta}} = - 2$

Hence, the value of $\frac{\gamma}{\alpha} +\bar{\frac{\alpha}{\beta}}$ is -2

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