11th Class Mathematics Sample Paper Mathematics Olympiad - Sample Paper-4

  • question_answer 9) If\[{{x}^{2}}+{{y}^{2}}=1\], then

    A) \[y.y''-2{{\left( y' \right)}^{2}}+1=0\]    

    B)  \[y.y''+{{(y')}^{2}}+1=0\]

    C) \[y.y''+{{\left( y' \right)}^{2}}-1=0\]                        

    D)  \[y.y''+2{{\left( y' \right)}^{2}}+1=0\]

    Correct Answer: B

    Solution :

    [b] \[\because {{x}^{2}}+y=1\] \[2x+2y.{{y}_{1}}=0\] \[{{y}_{1}}=-\frac{x}{y}\]                                  ..............(1) Again differentiating w.r.t. x, we have \[1+y_{1}^{2}+y.{{y}_{2}}=0\Rightarrow y.{{y}_{2}}+y_{1}^{2}+1=0\] \[\because {{y}_{1}}=\frac{dy}{dx}\,\,{{y}_{2}}=\frac{{{d}^{2}}y}{d{{x}^{2}}}\] Hence, option [b]is correct.

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