If α + β + γ = 0, then $$\frac{\text{α}^2}{\text{βγ}}+\frac{\text{β}^2}{\text{γα}}+\frac{\text{γ}^2}{\text{αβ}}=$$​ ?

Correct option is B

Given:

​$$\alpha + \beta + \gamma = 0$$​

We are to find the value of:

​$$\frac{\alpha^2}{\beta\gamma} + \frac{\beta^2}{\gamma\alpha} + \frac{\gamma^2}{\alpha\beta}$$​​

Formula Used:

$$\alpha^3 + \beta^3 + \gamma^3 - 3\alpha\beta\gamma = (\alpha + \beta + \gamma)\left(\alpha^2 + \beta^2 + \gamma^2 - \alpha\beta - \beta\gamma - \gamma\alpha\right)$$​

Since $$\alpha + \beta + \gamma = 0,$$​​

​$$\alpha^3 + \beta^3 + \gamma^3 = 3\alpha\beta\gamma$$

Solution:

​$$\frac{\alpha^2}{\beta\gamma} + \frac{\beta^2}{\gamma\alpha} + \frac{\gamma^2}{\alpha\beta}$$​​

​$$= \frac{\alpha^3 + \beta^3 + \gamma^3}{\alpha\beta\gamma}$$​​​​​

So,

​$$\frac{\alpha^3 + \beta^3 + \gamma^3}{\alpha\beta\gamma} = \frac{3\alpha\beta\gamma}{\alpha\beta\gamma} = 3$$​