$\mathrm{L}=\left(1,3\right), {\mathrm{L}}^1\left(1,-1\right)$ are the ends of latusrectum of a parabola. A is the vertex of the parabola then area of $∆{\mathrm{ALL}}^1$  in sq. units is

$\mathrm{L}=(1, 3) {\mathrm{L}}^1=(1, -1)$

Since  x-coordinates are equal then  parabola symmetric to x-axis

$$\begin{gathered} focus=(1, 1) \\ and 4a=4\Rightarrow \mathrm{a}=1 \end{gathered}$$

$$\begin{gathered} \Rightarrow \left(\mathrm{a}+\mathrm{h}, \mathrm{k}\right)=\left(1,1\right) \\ \Rightarrow \left(\mathrm{h}, \mathrm{k}\right)=\left(0, 1\right) \\ ∆=\frac{1}{2}\left|1(-2)+1(-2)+0\right| \\ =\frac{1}{2}\left|4\right|=2 \mathrm{sq}.units \end{gathered}$$