find the locus of the mid points of the chord of the hyperbola x2/a2 -y2/b2=1 which subtend a right angle at the origin.

consider the mid to be a point P (X_0,Y₀ ) now use the formula S₁=S₁₁  you will get the equation of chord with its mid point. then homogenise this equation with the equation of given hyperbola. since it subtends right andle at origin, from the homogenised equation equate coefficient of X²  and Y^2.