We know that for Rectangular Hyperbola, $b^{\prime}=a^{\prime}$ (the general form)

$\Rightarrow$ On simplifying Equation

1, $\frac{x^{2}}{\frac{4}{a-2}}-\frac{y^{2}}{\left(\frac{-4}{a}\right)}=1$

where we see, $a^{2}=\frac{4}{a-2}$ and $b^{2}=\frac{-4}{a}$

If $b^{\prime}=a^{\prime} \Rightarrow b^{\prime 2}=a^{\prime 2}$

$\Rightarrow \frac{-4}{a}=\frac{4}{a-2}$

$\Rightarrow-(a-2)=a$

$\Rightarrow 2=2 a$

$\Rightarrow a=1$