If the normals at $P, Q, R$ on the rectangular hyperbola  $xy=c^2$ intersect at a point S on the hyperbola, then  centroid of the triangle $PQR$ is at

detailed solution

Correct option is B

Normal at any point (ct,c/t) on the hyperbola xy=c2  if y−(c/t)=t2(x−ct) or t3x−ty+c−ct4=0 If it passes through another points (cα,c/α) on the hyperbola, then t3×cα−t×cα+c−ct4=0⇒t3α2−t+α−αt4=0⇒t3α+1(α−t)=0⇒t3α+1=0 as α≠twhich gives three values of t say t1,t2,t3 and hence three points P, Q, R on the hyperbola the normals at  which pass through S.We have t1+t2+t3=0=Σt1t2 and t1t2t3=−1/αCentroid of the triangle PQR isct1+t2+t33,c1/t1+1/t2+1/t33=(0,0)which is the centre of hyperbola.