Question

PQ and RS are two perpendicular chords of the rectangular hyperbola xy=c2. If C is the centre of the rectangular hyperbola, then the product of the slopes of CP,CQ,CR and CS is equal to:

Answer 1

Explanation:
Given:PQ and RS are two perpendicular chords of the rectangular hyperbola xy=c2 having centre C. We have to find the product of the slopes of CP,CQ,CR and CS.Let t1,t2,t3,t4 be the parameters of the point P,Q,R,S respectively.Then, the co-ordinates of P,Q,R,S are (ct1,ct1),(ct2,ct2),(ct3,ct3) and (ct4,ct4) respectively, which satisfy the given equation of hyperbola.Now, PQ⊥RS⇒ct2−ct1ct2−ct1×ct4−ct3ct4−ct3=−1[Product of slopes of perpendicular lines is equal to -1]⇒−(t2−t1)t1t2(t2−t1)×−(t4−t3)t3t4(t4−t3)=1t1t2×1t3t4=1⇒t1t2t3t4=1[equation (i)]Now, slope of CP=ct1−0ct1−0=1t12Similarly, we can find slopes of CQ,CR and CS are 1t22,1t32,1t42 respectively∴ Product of the slopes of CP,CQ,CR and CS =1t12×1t22×1t32×1t42=1t12t22t32t42=1[ using (i) ]Hence, the correct answer is 1.