$H: \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$

Foci : S (ae, 0), S' $(- ae , 0)$

Foot of directrix of parabola is $(- ae , 0)$

Focus of parabola is (ae, 0 )

Now, semi latus rectum of parabola $=\left| SS ^{\prime}\right|=2 ae$

Given, $4 a e = e \left(\frac{2 b ^{2}}{ a }\right)$

$b ^{2}=2 a ^{2}$

Given, $(2 \sqrt{2},-2 \sqrt{2})$ lies on $H$

$\frac{1}{ a ^{2}}-\frac{1}{ b ^{2}}=\frac{1}{8}$

From $(1)$ and $(2)$

$a^{2}=4, b^{2}=8$

$\because b^{2}=a^{2}\left(e^{2}-1\right)$

$\therefore e=\sqrt{3}$

Equation of parabola is $y^{2}=8 \sqrt{3} x$