Suppose, `a, b, c ge 0, c ne 1, a^(2) + b^(2) + c^(2) = c`. If `|(a+ib)/(2-c)| = (1)/(2)`, then c is equal to ____________.

Answer

To solve the problem step by step, we start with the given conditions and equations.

### Step 1: Write down the given conditions
We have:
1. \( a, b, c \geq 0 \)
2. \( c \neq 1 \)
3. \( a^2 + b^2 + c^2 = c \)
4. \( \left| \frac{a + ib}{2 - c} \right| = \frac{1}{2} \)

### Step 2: Analyze the modulus condition
From the modulus condition, we can express it as:
\[ \left| a + ib \right| = \frac{1}{2} |2 - c| \]
The modulus \( |a + ib| \) is given by:
\[ |a + ib| = \sqrt{a^2 + b^2} \]
Thus, we can rewrite the equation:
\[ \sqrt{a^2 + b^2} = \frac{1}{2} |2 - c| \]

### Step 3: Square both sides
Squaring both sides gives:
\[ a^2 + b^2 = \frac{(2 - c)^2}{4} \]

### Step 4: Substitute \( a^2 + b^2 \) in the equation
Now we can substitute \( a^2 + b^2 \) into the equation \( a^2 + b^2 + c^2 = c \):
\[ \frac{(2 - c)^2}{4} + c^2 = c \]

### Step 5: Clear the fraction
Multiply through by 4 to eliminate the fraction:
\[ (2 - c)^2 + 4c^2 = 4c \]

### Step 6: Expand and rearrange
Expanding \( (2 - c)^2 \):
\[ 4 - 4c + c^2 + 4c^2 = 4c \]
Combine like terms:
\[ 5c^2 - 8c + 4 = 0 \]

### Step 7: Solve the quadratic equation
Now we can solve for \( c \) using the quadratic formula:
\[ c = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 5, b = -8, c = 4 \):
\[ c = \frac{8 \pm \sqrt{(-8)^2 - 4 \cdot 5 \cdot 4}}{2 \cdot 5} \]
Calculating the discriminant:
\[ c = \frac{8 \pm \sqrt{64 - 80}}{10} = \frac{8 \pm \sqrt{-16}}{10} \]
This gives:
\[ c = \frac{8 \pm 4i}{10} = \frac{4 \pm 2i}{5} \]

### Step 8: Conclusion
Since \( c \) must be a real number and \( c \geq 0 \), the only possible value for \( c \) that satisfies the conditions is:
\[ c = \frac{4}{5} \]

Thus, the final answer is:
\[ \text{c is equal to } \frac{4}{5}. \]