A) \( \Large 3a^{4} \) |
B) \( \Large \frac{3}{4}a^{4} \) |
C) \( \Large 4a^{4} \) |
D) none of these |
B) \( \Large \frac{3}{4}a^{4} \) |
Since, the triangle is an equilateral triangle
Therefore, Area of equilateral triangle = \( \Large \frac{\sqrt{3}}{4}a^{2} \) ...(i)
Also, if \( \Large \left(x_{1},y_{1}\right),\ \left(x_{2},y_{2}\right)\ and\ \left(x_{3}, y_{3}\right) \) are the vertices of A, then
Area of equilateral triangle = \( \frac{1}{2} \begin{vmatrix}
x_{1} & y_{1} & 1 \\
x_{2} & y_{2} & 1 \\
x_{3} & y_{3} & 1
\end{vmatrix} \)...(ii)
From equations (i) and (ii) we get
\( \begin{vmatrix}
x_{1} & y_{1} & 1 \\
x_{2} & y_{2} & 1 \\
x_{3} & y_{3} & 1
\end{vmatrix} = \frac{\sqrt{3}}{2}a^{2} \)
= \( \Large
\begin{vmatrix}
x_{1} & y_{1} & 1 \\
x_{2} & y_{2} & 1 \\
x_{3} & y_{3} & 1
\end{vmatrix} = \left(\frac{\sqrt{3}}{2}a^{2}\right)^{2}= \frac{3a^{4}}{4} \)