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  4. Let A(x1, y1), B(x2, y2) and C(x3, y3) be vertices of an equilateral triangle whose side is 4 units. Let Δ=| x1 y1 1 x2 y2 1 x3 y3 1 | then Δ2 is equal to

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Q. Let $A\left(x_1, y_1\right), B\left(x_2, y_2\right)$ and $C\left(x_3, y_3\right)$ be vertices of an equilateral triangle whose side is 4 units. Let $\Delta=\begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix}$ then $\Delta^2$ is equal to

Determinants

Solution:

$\frac{1}{2}|\Delta|=\text { area of triangle }=\frac{\sqrt{3}}{4}(4)^2=4 \sqrt{3}$
$\Rightarrow|\Delta|=8 \sqrt{3} \Rightarrow \Delta^2=192$