what is rectangular hyperbola? also mention it's property.

the hyperbola for which the asymptotes are perpendicular also is called an equilateral hyperbola. this occurs when the semimajor and the semiminor axes are equal. this corresponds to take a=b,giving eccentricity e=√2 plugging a=b into general equation of hyperbola with semimajor axis parllel to x axis and semiminor axis parllel to y axis 

(x-x_o)÷a²-(y-y_o)÷b²=1

therefore it gives you (x-x_o)²-(y-y_o)²=a²

the rectangular hyperbola opening to the left has polar equation like this r²=a²sec(2θ)

and the rectangular hyperbola opening in the first and third quadrants has cartesian equation

xy=a²