KVPY Sample Paper KVPY Stream-SX Model Paper-8

A chord of the parabola \[y=-\,{{a}^{2}}{{x}^{2}}+5\,ax-\,4\] touches the curve \[y=\frac{1}{1-x}\] at the point \[x=2\]and is bisected by that point. If S is the sum of all possible values of a, then find 12 S:

A) 12

B) 15

C) 17

D) 19

Correct Answer: A

Solution :

Slope of the tangent \[{{\left. \frac{dy}{dx}=\frac{1}{{{(1-x)}^{2}}} \right|}_{x=2}}=1\]

Equation of tangent is

\[y+1=1\,(x-2)\] i.e. \[y=x-3\]

Parabola \[y=-\,{{a}^{2}}{{x}^{2}}+5ax-4\]

Solving the equations of tangent and the parabola

\[x-3=-{{a}^{2}}{{x}^{2}}+5\,\,ax-4\]

\[{{a}^{2}}{{x}^{2}}+(1-5\,a)\,x+1=0\]

Since x is real \[\therefore {{(1-5a)}^{2}}-4{{a}^{2}}\ge 0\]

\[\Rightarrow a\le \frac{1}{7}\,\,or\,\,a\ge \frac{1}{3}\]

Sum of roots \[=\frac{5a-1}{{{a}^{2}}}=2\times 2\]

\[4{{a}^{2}}-5a+1=0\] \[a=1,\frac{1}{4}\] [\[[a=\frac{1}{4}\]is rejected ]

\[\therefore \,\,\,S=1\] \[\therefore \,\,\,12\,S=12\]