The locus of the foot of the perpendicular from the center  of the hyperbola xy = 1 on a variable tangent is

Let the foot of perpendicular from O(0,0) to the tangent to the hyperbola be P(h, k). Slope of OP=khThen the equation of tangent to the hyperbola is y−k=−hk(x−h)or hx+ky=h2+k2Solving it with xy = 1, we have hx+kx=h2+k2or  hx2−h2+k2x+k=0This equation must have real and equal roots. HenceD=0or  h2+k22−4hk=0or  x2+y22=4xy