If `A (x_(1), y_(1)), B (x_(2), y_(2)) and C (x_(3), y_(3))` are vertices of an equilateral triangle whose each side is equal to a, then prove that
`|(x_(1),y_(1),2),(x_(2),y_(2),2),(x_(3),y_(3),2)|` is equal to

Options

A. `2a^(2)`
B. `2a^(4)`
C. `3a^(2)`
D. `3a^(4)`

Explanation

To prove that the determinant \( |(x_1, y_1, 2), (x_2, y_2, 2), (x_3, y_3, 2)| \) is equal to \( 3a^4 \) for the vertices of an equilateral triangle with each side equal to \( a \), we can follow these steps:

### Step 1: Area of Triangle
The area \( \Delta \) of triangle \( ABC \) with vertices \( A(x_1, y_1) \), \( B(x_2, y_2) \), and \( C(x_3, y_3) \) can be expressed using the determinant:
\[ \Delta = \frac{1}{2} \left| \begin{array}{ccc} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{array} \right| \]

### Step 2: Multiply by 2
Multiplying both sides by 2 gives:
\[ 2\Delta = \left| \begin{array}{ccc} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{array} \right| \]

### Step 3: Multiply by 2 Again
Now, we multiply the determinant by 2 in the third column:
\[ 4\Delta = \left| \begin{array}{ccc} x_1 & y_1 & 2 \\ x_2 & y_2 & 2 \\ x_3 & y_3 & 2 \end{array} \right| \]

### Step 4: Area of Equilateral Triangle
The area \( \Delta \) of an equilateral triangle with side length \( a \) is given by:
\[ \Delta = \frac{\sqrt{3}}{4} a^2 \]

### Step 5: Substitute Area into the Equation
Substituting the area into the equation from Step 3:
\[ 4 \left( \frac{\sqrt{3}}{4} a^2 \right) = \left| \begin{array}{ccc} x_1 & y_1 & 2 \\ x_2 & y_2 & 2 \\ x_3 & y_3 & 2 \end{array} \right| \]
This simplifies to:
\[ \sqrt{3} a^2 = \left| \begin{array}{ccc} x_1 & y_1 & 2 \\ x_2 & y_2 & 2 \\ x_3 & y_3 & 2 \end{array} \right| \]

### Step 6: Square Both Sides
Squaring both sides gives:
\[ 3a^4 = \left| \begin{array}{ccc} x_1 & y_1 & 2 \\ x_2 & y_2 & 2 \\ x_3 & y_3 & 2 \end{array} \right|^2 \]

### Conclusion
Thus, we have shown that:
\[ \left| \begin{array}{ccc} x_1 & y_1 & 2 \\ x_2 & y_2 & 2 \\ x_3 & y_3 & 2 \end{array} \right| = \sqrt{3} a^2 \]
and squaring both sides leads to:
\[ \left| \begin{array}{ccc} x_1 & y_1 & 2 \\ x_2 & y_2 & 2 \\ x_3 & y_3 & 2 \end{array} \right|^2 = 3a^4 \]