If the least and the largest real values of `alpha`, for which the equation `z + alpha |z – 1| + 2i = 0` (z `in` C and 1 =`sqrt(-1)` ) has a solution, are p and q respectively, then `4(p^2 + q^2)` is equal to ____

Answer

To solve the equation \( z + \alpha |z - 1| + 2i = 0 \) for the values of \( \alpha \), we will follow these steps:

### Step 1: Rewrite the equation
Let \( z = x + iy \), where \( x \) and \( y \) are real numbers. Then, we can rewrite the equation as:
\[ x + iy + \alpha |(x + iy) - 1| + 2i = 0 \]
This simplifies to:
\[ x + \alpha |(x - 1) + iy| + (y + 2)i = 0 \]

### Step 2: Separate real and imaginary parts
From the equation, we can separate the real and imaginary parts:
- Real part: \( x + \alpha |(x - 1) + iy| = 0 \)
- Imaginary part: \( y + 2 = 0 \)

### Step 3: Solve for \( y \)
From the imaginary part, we find:
\[ y + 2 = 0 \implies y = -2 \]

### Step 4: Substitute \( y \) into the real part
Substituting \( y = -2 \) into the real part gives:
\[ x + \alpha |(x - 1) - 2i| = 0 \]
Now, we need to compute \( |(x - 1) - 2i| \):
\[ |(x - 1) - 2i| = \sqrt{(x - 1)^2 + (-2)^2} = \sqrt{(x - 1)^2 + 4} \]
Thus, the equation becomes:
\[ x + \alpha \sqrt{(x - 1)^2 + 4} = 0 \]

### Step 5: Rearranging the equation
Rearranging gives:
\[ \alpha \sqrt{(x - 1)^2 + 4} = -x \]
This implies:
\[ \alpha = -\frac{x}{\sqrt{(x - 1)^2 + 4}} \]

### Step 6: Determine the range of \( \alpha \)
To find the least and largest values of \( \alpha \), we analyze the function:
\[ f(x) = -\frac{x}{\sqrt{(x - 1)^2 + 4}} \]
We will find the maximum and minimum values of \( f(x) \).

### Step 7: Find critical points
To find the critical points, we differentiate \( f(x) \) and set the derivative to zero. This involves using the quotient rule:
Let \( u = -x \) and \( v = \sqrt{(x - 1)^2 + 4} \):
\[ f'(x) = \frac{u'v - uv'}{v^2} \]
where \( u' = -1 \) and \( v' = \frac{(x - 1)}{\sqrt{(x - 1)^2 + 4}} \).

Setting \( f'(x) = 0 \) leads to solving for \( x \).

### Step 8: Analyze the limits
We also need to check the limits as \( x \to \pm \infty \) to find the bounds for \( \alpha \).

### Step 9: Calculate \( p \) and \( q \)
After finding the critical points and limits, we can determine the least value \( p \) and the largest value \( q \) of \( \alpha \).

### Step 10: Calculate \( 4(p^2 + q^2) \)
Finally, we compute:
\[ 4(p^2 + q^2) \]