Hyperbola- rectangular hyperbola (theory complete)

Find all values for xy=c^(2) (assymptotes; transverse axis; conjugate axis; vertex; focii length of latus rectum; Equation of auxilliary circle; equation of directrix circle and hyperbola and rectangular hyperbola intersects at right angle.

Show that for rectangular hyperbola xy=c^(2)

A triangle has its vertices on a rectangular hyperbola.Prove that the orthocentre of the triangle also lies on the same hyperbola.

A triangle has its vertices on a rectangular hyperbola.Prove that the orthocenter of the triangle also lies on the same hyperbola.

The differential equation of rectangular hyperbolas whose axes are asymptotes of the hyperbola x^2 - y^2= a^2 , is :

Normal to a rectangular hyperbola at P meets the transverse axis at N. If foci of hyperbola are S and S', then find the value of (SN)/(SP).