For a complex number Z, if one root of the equation `Z^(2)-aZ+a=0` is `(1+i)` and its other root is `alpha`, then the value of `(a)/(alpha^(4))` is equal to

Options

A. 4
B. `-(1)/(4)`
C. 2
D. `-2`

Answer

To solve the problem, we will follow these steps:

### Step 1: Identify the roots and coefficients of the quadratic equation
The given quadratic equation is:
\[ Z^2 - aZ + a = 0 \]
We know that one root is \( Z_1 = 1 + i \) and the other root is \( Z_2 = \alpha \).

### Step 2: Use Vieta's formulas
According to Vieta's formulas:
- The sum of the roots \( Z_1 + Z_2 = a \)
- The product of the roots \( Z_1 \cdot Z_2 = a \)

From the first formula:
\[ (1 + i) + \alpha = a \]
This can be rearranged to find \( \alpha \):
\[ \alpha = a - (1 + i) \]

From the second formula:
\[ (1 + i) \cdot \alpha = a \]

### Step 3: Substitute \( \alpha \) into the product equation
Substituting \( \alpha \) from the first equation into the second:
\[ (1 + i)(a - (1 + i)) = a \]
Expanding this:
\[ (1 + i)(a - 1 - i) = a \]
\[ (1 + i)(a - 1 - i) = (1 + i)(a - 1) - (1 + i)i \]
\[ = (a - 1) + i(a - 1) - (i + 1) \]
Combining like terms:
\[ = (a - 2) + i(a - 2) \]
Setting this equal to \( a \):
\[ (a - 2) + i(a - 2) = a \]

### Step 4: Separate real and imaginary parts
From the equation:
- Real part: \( a - 2 = a \) (which gives no new information)
- Imaginary part: \( a - 2 = 0 \) (which gives \( a = 2 \))

### Step 5: Find \( \alpha \)
Now substituting \( a = 2 \) back into the equation for \( \alpha \):
\[ \alpha = 2 - (1 + i) = 1 - i \]

### Step 6: Calculate \( \alpha^4 \)
Now we need to compute \( \alpha^4 \):
\[ \alpha = 1 - i \]
To find \( \alpha^2 \):
\[ \alpha^2 = (1 - i)^2 = 1 - 2i + i^2 = 1 - 2i - 1 = -2i \]
Now, squaring \( \alpha^2 \) to find \( \alpha^4 \):
\[ \alpha^4 = (-2i)^2 = 4(-1) = -4 \]

### Step 7: Calculate \( \frac{a}{\alpha^4} \)
Now we can find:
\[ \frac{a}{\alpha^4} = \frac{2}{-4} = -\frac{1}{2} \]

### Final Answer
Thus, the value of \( \frac{a}{\alpha^4} \) is:
\[ \boxed{-\frac{1}{2}} \]