A tangent at any point P on the rectangular hyperbola xy=c2  meets the lines y = x and y = - x at Q and R respectively. The normal at P meets X and Y axes at M and N respectively. If  A1,A2 be the  areas of triangles OQR and OMN respectively, then which of the following is/are not independent of P?

P=ct,ct tangent is x+yt2=2ct→1 Normal is t3x−ty=ct4−1→2solve 1 with the line y=x , we get Q and with the line y=-x, we get R Q=2ct1+t2,2ct1+t2,R=2ct1−t2,−2ct1−t2,;now solve 2 with the line y=0 ,we get R and with the line x=0, we get NM=ct4−1t3,0  N=0,c1-t4tArea of triangle OQR=A1=4c2t21−t4, Area of triangle OMN=A2=c21−t422t4,      since area of triangle formed by the vertices 0,0,x1,y1,x2,y2 is =12x1y2-x2y1now A12A2=8c6, which is independent of P