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JAMIA MILLIA ISLAMIA Jamia Millia Islamia Solved Paper-2011

  • question_answer
        If tangent to the curve\[{{y}^{2}}={{x}^{3}}\]at its point \[({{m}^{2}},{{m}^{3}})\]is also normal to the curve at \[({{M}^{2}},\text{ }{{M}^{3}}),\]then what is the value of\[mM\]?

    A)  \[-\frac{4}{9}\]                                

    B)  \[-\frac{2}{9}\]

    C)  \[-\frac{1}{3}\]                                

    D)  1

    Correct Answer: A

    Solution :

                    Given equation of curve is\[{{y}^{2}}={{x}^{3}}\] \[\Rightarrow \]               \[2y=\frac{dy}{dx}=3{{x}^{2}}\] \[\Rightarrow \]               \[\frac{dy}{dx}=\frac{3{{x}^{2}}}{2y}\] \[\therefore \]  \[{{\frac{dy}{dx}}_{({{m}^{2}},{{m}^{3}})}}=\frac{3{{m}^{4}}}{2{{m}^{3}}}=\frac{3}{2}m\] and        \[{{\frac{dy}{dx}}_{({{M}^{2}},{{M}^{3}})}}=\frac{3{{m}^{4}}}{2{{m}^{3}}}=\frac{3}{2}m\] Also,      \[\frac{3}{2}m\times \frac{3}{2}M=-1\] \[\Rightarrow \]               \[mM=-\frac{4}{9}\]


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