KVPY Sample Paper KVPY Stream-SX Model Paper-22

A chord of the parabola \[y=-\,{{a}^{2}}{{x}^{2}}+5ax-4\] 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 12S.

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

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

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

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]

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