Let `4x^(2)-4(alpha-2)x + alpha-2=0(alpha in R)` be a quadratic equation. Find the values of `alpha` for which
Both the roots are equal in magnitude but opposite in sign

Answer

To solve the problem, we need to find the values of \( \alpha \) for which the quadratic equation \( 4x^2 - 4(\alpha - 2)x + (\alpha - 2) = 0 \) has roots that are equal in magnitude but opposite in sign.

### Step 1: Identify the coefficients
The given quadratic equation is in the standard form \( ax^2 + bx + c = 0 \), where:
- \( a = 4 \)
- \( b = -4(\alpha - 2) \)
- \( c = \alpha - 2 \)

### Step 2: Condition for equal magnitude and opposite sign
For the roots to be equal in magnitude but opposite in sign, the coefficient \( b \) must be zero. This is because if \( b = 0 \), the quadratic simplifies to \( ax^2 + c = 0 \), leading to roots of the form \( x = \pm \sqrt{-\frac{c}{a}} \).

### Step 3: Set \( b = 0 \)
Set the coefficient \( b \) equal to zero:
\[ -4(\alpha - 2) = 0 \]
Solving this gives:
\[ \alpha - 2 = 0 \quad \Rightarrow \quad \alpha = 2 \]

### Step 4: Check the condition for \( c \)
Next, we need to ensure that \( c \) is negative (since we want the roots to be real and opposite in sign):
\[ c = \alpha - 2 \]
Substituting \( \alpha = 2 \):
\[ c = 2 - 2 = 0 \]
Since \( c = 0 \) does not satisfy the condition for having roots that are real and opposite in sign, we need to check if there are any other values of \( \alpha \) that satisfy the conditions.

### Step 5: Analyze the sign of \( c \)
For the roots to be real, we need:
\[ c < 0 \quad \Rightarrow \quad \alpha - 2 < 0 \quad \Rightarrow \quad \alpha < 2 \]

### Step 6: Conclusion
From the above steps, we have:
1. \( \alpha = 2 \) gives us roots that are equal in magnitude but not opposite in sign.
2. \( \alpha < 2 \) does not yield roots that are equal in magnitude and opposite in sign.

Thus, there are no values of \( \alpha \) for which the quadratic equation has roots that are equal in magnitude and opposite in sign.

### Final Answer
There is no value of \( \alpha \) for which the quadratic equation has roots that are equal in magnitude and opposite in sign.