Statement-1 If a tangent to the hyperbola  $2x^2-4y^2=8$ meets x-axis at $P$ and $y$-axis at Q. Lines PR and  $QR$ are drawn such that $OPRQ$is a rectangle where $O$ is the origin then locus of $R$ is a hyperbola. Statement-2 The curve described parametrically by 

$x=\frac{2}{3}+4t\text{ and }y=-\frac{4}{7}+\frac{4}{t}$ represents a hyperbola.

Tangent to the hyperbola $\frac{x^2}{4}-\frac{y^2}{2}=1$ at a point $\left(x_1,y_1\right)\text{ is }\frac{x{x}_1}{4}-\frac{y{y}_1}{2}=1$

Coordinates of $P$ are $\left(\frac{4}{x_1},0\right)\text{ of }Q\text{ are }\left(0,\frac{-2}{y_1}\right)$ and of $R$ are 

$\left(\frac{4}{x_1},\frac{-2}{y_1}\right)=(h,k)$

Also $\frac{x_1^2}{4}-\frac{y_1^2}{2}=1\text{ as }\left(x_1,y_1\right)$ as $\left(x_1,y_1\right)$ lies on the hyperbola

$\Rightarrow \frac{4}{h^2}-\frac{2}{k^2}=1$

Locus of $(h,k)\text{ is }\frac{4}{x^2}-\frac{2}{y^2}=1$

Which is not a hyperbola and thus statemnet-1 is false.

In statement-2 we have $\left(x-\frac{2}{3}\right)\left(y+\frac{4}{7}\right)=16$ which represents a rectangular hyperbola and thus is true.