Explanation

(a)
The given equation may be written as
$\frac{x^{2}}{\frac{32}{3}} - \frac{y^{2}}{8} = 1$
$\Rightarrow \ \ \frac{x^{2}}{\left( \frac{4\sqrt{2}}{\sqrt{3}} \right)^{2}} - \frac{y^{2}}{\left( 2\sqrt{2} \right)^{2}} = 1$
On comparing the given equation with the standard equation $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1,$ we get
$a^{2} = \left( \frac{4\sqrt{2}}{\sqrt{3}} \right)^{2}$ and $b^{2} = \left( 2\sqrt{2} \right)^{2}$
$\therefore\$ Length of transverse axis of a hyperbola
$= 2a = 2 \times \frac{4\sqrt{2}}{\sqrt{3}} = \frac{8\sqrt{2}}{\sqrt{3}}$