Suppose `x+iy=|(7i,-5i,1),(14,5i,-1),(28,5,i)|` then `sqrt((x+1//4)^(2)+y^(2))=`__________

Answer

To solve the problem, we need to evaluate the determinant given in the question and then find the required expression. Let's break it down step-by-step.

### Step 1: Evaluate the Determinant

We have the determinant:

\[ D = \begin{vmatrix} 7i & -5i & 1 \\ 14 & 5i & -1 \\ 28 & 5 & i \end{vmatrix} \]

Using the determinant formula for a 3x3 matrix, we can expand it as follows:

\[ D = 7i \begin{vmatrix} 5i & -1 \\ 5 & i \end{vmatrix} - (-5i) \begin{vmatrix} 14 & -1 \\ 28 & i \end{vmatrix} + 1 \begin{vmatrix} 14 & 5i \\ 28 & 5 \end{vmatrix} \]

### Step 2: Calculate the 2x2 Determinants

1. **First 2x2 Determinant:**

\[ \begin{vmatrix} 5i & -1 \\ 5 & i \end{vmatrix} = (5i)(i) - (-1)(5) = 5i^2 + 5 = 5(-1) + 5 = -5 + 5 = 0 \]

2. **Second 2x2 Determinant:**

\[ \begin{vmatrix} 14 & -1 \\ 28 & i \end{vmatrix} = (14)(i) - (-1)(28) = 14i + 28 \]

3. **Third 2x2 Determinant:**

\[ \begin{vmatrix} 14 & 5i \\ 28 & 5 \end{vmatrix} = (14)(5) - (5i)(28) = 70 - 140i \]

### Step 3: Substitute Back into the Determinant Expression

Now substituting back into the determinant expression:

\[ D = 7i(0) + 5i(14i + 28) + (70 - 140i) \]

This simplifies to:

\[ D = 0 + 5i(14i + 28) + (70 - 140i) \]

Calculating \(5i(14i + 28)\):

\[ = 5i(14i) + 5i(28) = 70i^2 + 140i = 70(-1) + 140i = -70 + 140i \]

Now substituting this back into \(D\):

\[ D = (-70 + 140i) + (70 - 140i) = 0 \]

### Step 4: Finding \(x\) and \(y\)

From the determinant, we have:

\[ x + iy = 0 \implies x = 0 \text{ and } y = 0 \]

### Step 5: Calculate the Required Expression

Now we need to find:

\[ \sqrt{\left(x + \frac{1}{4}\right)^2 + y^2} \]

Substituting \(x = 0\) and \(y = 0\):

\[ = \sqrt{\left(0 + \frac{1}{4}\right)^2 + 0^2} = \sqrt{\left(\frac{1}{4}\right)^2} = \sqrt{\frac{1}{16}} = \frac{1}{4} \]

### Final Answer

Thus, the final answer is:

\[ \frac{1}{4} = 0.25 \]