What is the area of the triangle with vertices `(x_(1) , (1)/(x_(1))) * (x_(2) , (1)/(x_(2))) * (x_(3) , (1)/(x_(3))) ?`

Options

A. A. `|(x_(1) - x_(2)) (x_(2) - x_(3)) (x_(3) - x_(1))|`
B. B. 0
C. C. `|((x_(1) - x_(2)) (x_(2) - x_(3)) (x_(3) - x_(1)))/(x_(1) x_(2) x_(3))|`
D. D. `|((x_(1) - x_(2)) (x_(2) - x_(3)) (x_(3) - x_(1)))/(2 x_(1) x_(2) x_(3))|`

Answer

To find the area of the triangle with vertices \((x_1, \frac{1}{x_1})\), \((x_2, \frac{1}{x_2})\), and \((x_3, \frac{1}{x_3})\), we can use the formula for the area of a triangle given by the coordinates of its vertices. The formula is:

\[ \text{Area} = \frac{1}{2} \left| \begin{vmatrix} x_1 & \frac{1}{x_1} & 1 \\ x_2 & \frac{1}{x_2} & 1 \\ x_3 & \frac{1}{x_3} & 1 \end{vmatrix} \right| \]

### Step 1: Set up the determinant
We will calculate the determinant of the matrix formed by the coordinates of the vertices:

\[ \begin{vmatrix} x_1 & \frac{1}{x_1} & 1 \\ x_2 & \frac{1}{x_2} & 1 \\ x_3 & \frac{1}{x_3} & 1 \end{vmatrix} \]

### Step 2: Expand the determinant
Using the determinant expansion method, we can expand this determinant along the first column:

\[ = x_1 \begin{vmatrix} \frac{1}{x_2} & 1 \\ \frac{1}{x_3} & 1 \end{vmatrix} - x_2 \begin{vmatrix} \frac{1}{x_1} & 1 \\ \frac{1}{x_3} & 1 \end{vmatrix} + x_3 \begin{vmatrix} \frac{1}{x_1} & 1 \\ \frac{1}{x_2} & 1 \end{vmatrix} \]

### Step 3: Calculate the 2x2 determinants
Now we calculate each of the 2x2 determinants:

1. For the first determinant:
\[ \begin{vmatrix} \frac{1}{x_2} & 1 \\ \frac{1}{x_3} & 1 \end{vmatrix} = \frac{1}{x_2} \cdot 1 - 1 \cdot \frac{1}{x_3} = \frac{1}{x_2} - \frac{1}{x_3} = \frac{x_3 - x_2}{x_2 x_3} \]

2. For the second determinant:
\[ \begin{vmatrix} \frac{1}{x_1} & 1 \\ \frac{1}{x_3} & 1 \end{vmatrix} = \frac{1}{x_1} \cdot 1 - 1 \cdot \frac{1}{x_3} = \frac{1}{x_1} - \frac{1}{x_3} = \frac{x_3 - x_1}{x_1 x_3} \]

3. For the third determinant:
\[ \begin{vmatrix} \frac{1}{x_1} & 1 \\ \frac{1}{x_2} & 1 \end{vmatrix} = \frac{1}{x_1} \cdot 1 - 1 \cdot \frac{1}{x_2} = \frac{1}{x_1} - \frac{1}{x_2} = \frac{x_2 - x_1}{x_1 x_2} \]

### Step 4: Substitute back into the determinant
Substituting these back into our expression for the determinant, we have:

\[ = x_1 \cdot \frac{x_3 - x_2}{x_2 x_3} - x_2 \cdot \frac{x_3 - x_1}{x_1 x_3} + x_3 \cdot \frac{x_2 - x_1}{x_1 x_2} \]

### Step 5: Combine and simplify
Combining these terms under a common denominator \(x_1 x_2 x_3\):

\[ = \frac{x_1^2 (x_3 - x_2) - x_2^2 (x_3 - x_1) + x_3^2 (x_2 - x_1)}{x_1 x_2 x_3} \]

### Step 6: Calculate the area
Now, the area of the triangle is:

\[ \text{Area} = \frac{1}{2} \left| \frac{x_1^2 (x_3 - x_2) - x_2^2 (x_3 - x_1) + x_3^2 (x_2 - x_1)}{x_1 x_2 x_3} \right| \]

### Final Result
Thus, the area of the triangle is:

\[ \text{Area} = \frac{1}{2} \cdot \frac{|x_1^2 (x_3 - x_2) - x_2^2 (x_3 - x_1) + x_3^2 (x_2 - x_1)|}{x_1 x_2 x_3} \]