If \( A(x_1,y_1),\ B(x_2,y_2) \) and \( C(x_3,y_3) \) are the vertices of an equilateral triangle whose each side is equal to \( k \), then the value of \( \begin{vmatrix}x_1 & y_1 & 2\\x_2 & y_2 & 2\\x_3 & y_3 & 2\end{vmatrix}^2 \) is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \( 3k^4 \)
The area of a triangle with these vertices is
\( \displaystyle \Delta=\frac{1}{2}\left|\begin{vmatrix}x_1 & y_1 & 1\\x_2 & y_2 & 1\\x_3 & y_3 & 1\end{vmatrix}\right| \)
Taking the common factor 2 out of the third column:
\( \displaystyle \begin{vmatrix}x_1 & y_1 & 2\\x_2 & y_2 & 2\\x_3 & y_3 & 2\end{vmatrix}=2\begin{vmatrix}x_1 & y_1 & 1\\x_2 & y_2 & 1\\x_3 & y_3 & 1\end{vmatrix} \)
\( \displaystyle =2(\pm 2\Delta)=\pm 4\Delta \)
For an equilateral triangle of side k:
\( \displaystyle \Delta=\frac{\sqrt{3}}{4}k^2 \)
\( \displaystyle \begin{vmatrix}x_1 & y_1 & 2\\x_2 & y_2 & 2\\x_3 & y_3 & 2\end{vmatrix}^2=16\Delta^2 \)
\( \displaystyle =16\times\frac{3}{16}k^4 \)
\( \displaystyle =3k^4 \)