KVPY Sample Paper KVPY Stream-SX Model Paper-20

If a curve passes through the point \[(1,-2)\] and has slope of the tangent at any point \[(x,y)\] on it as \[\frac{{{x}^{2}}-2y}{x},\] then the curve also passes through the point:

A) \[(3,0)\]

B) \[(\sqrt{3,}0)\]

C) \[(-1,\,\,2)\]

D) \[\left( -\sqrt{2},1 \right)\]

Correct Answer: B

Solution :

\[\frac{dy}{dx}=\frac{{{x}^{2}}-2y}{x}\] (Given)

\[\frac{dy}{dx}+2\frac{y}{x}=x\]

I.F \[={{e}^{\int{\frac{2}{x}dx}}}={{x}^{2}}\]

\[\therefore \] \[y.{{x}^{2}}=\int{x.{{x}^{2}}}dx+C\]

\[y.{{x}^{2}}=\frac{{{x}^{4}}}{y}+C\]

Hence, b passes through\[(1,-2)\]

\[\Rightarrow \] \[C=-\frac{9}{4}\]

\[\therefore \] \[y{{x}^{2}}=\frac{{{x}^{4}}}{4}-\frac{9}{4}\]

Checking options, only option [B] satisfy