• If the area of a triangle is given, then both positive and negative values of the determinant are used for calculation.why?

Determinant can have both positive and negative values, but area is always positive.
Consider a simple example:

Let us find the coordinate of  a missing point of a triangle, when the area is known.
Let A(-2,2);B(1,5) and C(x,-1) be the three coordinates and area of the triangle =16.5
$$\begin{gathered} \mathrm{Now}, \\ \mathrm{Area} =\pm \frac{1}{2}\left|\begin{array}{ccc}-2& 2& 1\\ 1& 5& 1\\ \mathrm{x}& -1& 1\end{array}\right| \\ \Rightarrow 16.5=\pm \frac{1}{2}[-2(5+1)-1(2+1)+\mathrm{x}(2-5\left)\right] \\ \Rightarrow 33=\pm [-15-3\mathrm{x}] \\ \Rightarrow 11=\pm [-5-\mathrm{x}] \\ \mathrm{Now}, \\ -5-\mathrm{x}=11 \mathrm{or} 5+\mathrm{x}=11 \\ \Rightarrow \mathrm{x}=-16 \mathrm{or} \mathrm{x}=6 \\ \mathrm{Now}, \mathrm{x}=-16 \mathrm{is} \mathrm{rejected} \mathrm{as} \mathrm{it} \mathrm{is} \mathrm{giving} \mathrm{wrong} \mathrm{area} \\ \mathrm{So}, \mathrm{we} \mathrm{obtain} \mathrm{x}=6 ,\mathrm{by} \mathrm{taking} \mathrm{negative} \mathrm{sign} \end{gathered}$$