If ` alpha, beta, gamma` and a, b, c are complex numbers such that `(alpha)/( a) + (beta)/( b) + (gamma)/(c) = 1 + i and (a)/( alpha ) +(b)/( beta) + (c)/( gamma) 0 ` , then the value of `(alpha^(2))/( a^(2)) + (beta^(2))/(b^(2)) + (gamma^(2))/( c^(2))` is equal to

Options

A. 0
B. `-1`
C. 2i
D. `-2i`

Answer

To solve the given problem, we will follow a systematic approach.

### Given:
1. \(\frac{\alpha}{a} + \frac{\beta}{b} + \frac{\gamma}{c} = 1 + i\)
2. \(\frac{a}{\alpha} + \frac{b}{\beta} + \frac{c}{\gamma} = 0\)

We need to find the value of:
\[ \frac{\alpha^2}{a^2} + \frac{\beta^2}{b^2} + \frac{\gamma^2}{c^2} \]

### Step 1: Square the first equation
Start by squaring the first equation:
\[ \left(\frac{\alpha}{a} + \frac{\beta}{b} + \frac{\gamma}{c}\right)^2 = (1 + i)^2 \]

Calculating the right-hand side:
\[ (1 + i)^2 = 1^2 + 2(1)(i) + i^2 = 1 + 2i - 1 = 2i \]

### Step 2: Expand the left-hand side
Now, expand the left-hand side:
\[ \left(\frac{\alpha}{a} + \frac{\beta}{b} + \frac{\gamma}{c}\right)^2 = \frac{\alpha^2}{a^2} + \frac{\beta^2}{b^2} + \frac{\gamma^2}{c^2} + 2\left(\frac{\alpha \beta}{ab} + \frac{\beta \gamma}{bc} + \frac{\gamma \alpha}{ca}\right) \]

### Step 3: Set up the equation
We now have:
\[ \frac{\alpha^2}{a^2} + \frac{\beta^2}{b^2} + \frac{\gamma^2}{c^2} + 2\left(\frac{\alpha \beta}{ab} + \frac{\beta \gamma}{bc} + \frac{\gamma \alpha}{ca}\right) = 2i \]

### Step 4: Use the second equation
From the second equation:
\[ \frac{a}{\alpha} + \frac{b}{\beta} + \frac{c}{\gamma} = 0 \]
This implies:
\[ \frac{b}{\beta} + \frac{c}{\gamma} = -\frac{a}{\alpha} \]

Multiplying through by \(\alpha \beta \gamma\) gives:
\[ b \gamma + c \alpha = -a \beta \]
Rearranging leads to:
\[ a \beta + b \gamma + c \alpha = 0 \]

### Step 5: Substitute back into the equation
Now, substituting this back into our earlier expression for \(t\):
Let \(t = \frac{\alpha \beta}{ab} + \frac{\beta \gamma}{bc} + \frac{\gamma \alpha}{ca}\).

Using the equation \(a \beta + b \gamma + c \alpha = 0\), we find that:
\[ t = 0 \]

### Step 6: Final calculation
Substituting \(t = 0\) back into our equation:
\[ \frac{\alpha^2}{a^2} + \frac{\beta^2}{b^2} + \frac{\gamma^2}{c^2} + 2(0) = 2i \]
Thus:
\[ \frac{\alpha^2}{a^2} + \frac{\beta^2}{b^2} + \frac{\gamma^2}{c^2} = 2i \]

### Conclusion
The value of \(\frac{\alpha^2}{a^2} + \frac{\beta^2}{b^2} + \frac{\gamma^2}{c^2}\) is:
\[ \boxed{2i} \]