Step 1:

The centre of the rectangular hyperbola is at $(-\large\frac{1}{2}.-\large\frac{1}{2})$ with asymptotes parallel to the coordinate axes.

Therefore its equation is of the form $(x+\large\frac{1}{2})($$y+\large\frac{1}{2})=$$c^2$

Step 2:

The point $(1,\large\frac{1}{4})$ lies on the rectangular hyperbola.

Therefore $(1+\large\frac{1}{2})(\large\frac{1}{4}+\frac{1}{2})=$$c^2$

$\Rightarrow c^2=\large\frac{9}{8}$

The equation of the rectangular hyperbola is $(x+\large\frac{1}{2})($$y+\large\frac{1}{2})=$$\large\frac{9}{8}$