KVPY Sample Paper KVPY Stream-SX Model Paper-8

If the conies whose equations are \[S\equiv {{\sin }^{2}}\theta {{x}^{2}}+2hxy+{{\cos }^{2}}\theta {{y}^{2}}+32x+16y+19=0,\]\[S\equiv {{\cos }^{2}}\theta {{x}^{2}}+2h'xy+{{\sin }^{2}}\theta {{y}^{2}}+16x+32y+19=0\] inersects in four concyclic points then, (where\[\theta \in R\])-

A) \[h+h'=0\]

B) \[h=h'\]

C) \[h+h'=1\]

D) None of these

Correct Answer: A

Solution :

\[S\equiv (si{{n}^{2}}\theta )\,\,{{x}^{2}}+2hxy+({{\cos }^{2}}\theta )\,\,{{y}^{2}}+32x+16y+19=0\]	??...(1)
\[S'\equiv ({{\cos }^{2}}\theta )\,\,{{x}^{2}}+2h'xy+({{\sin }^{2}}\theta )\,\,{{y}^{2}}+16x+32y+19=0\]	???(2)
\[\because \]Curve passing through point of intersection of S & S? is \[S+\lambda S'=0\]
\[\Rightarrow {{x}^{2}}({{\sin }^{2}}\theta +\lambda {{\cos }^{2}}\theta )+{{y}^{2}}({{\cos }^{2}}\theta +\lambda {{\sin }^{2}}\theta )+\]
\[2xy\,\,(h+\lambda h')+x\,\,(32+16\lambda )+y\,\,(16+32\lambda )+19\,\,(1+\lambda )=0\]For this equation to be a circle, then
\[{{\sin }^{2}}\theta +\lambda {{\cos }^{2}}\theta ={{\cos }^{2}}\theta +\lambda {{\sin }^{2}}\theta \]\[\Rightarrow \lambda =1\] and \[h+\lambda h'=0\]\[\Rightarrow h+h'=0\]