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JEE Main & Advanced Mathematics Applications of Derivatives Question Bank Critical Thinking

  • question_answer
    If \[y=4x-5\] is tangent to the curve \[{{y}^{2}}=p{{x}^{3}}+q\] at (2, 3), then [IIT 1994; UPSEAT 2001]

    A) \[p=2,q=-7\]

    B) \[p=-2,q=7\]

    C) \[p=-2,q=-7\]

    D) \[p=2,q=7\]

    Correct Answer: A

    Solution :

    • Given curve \[{{y}^{2}}=p{{x}^{3}}+q\]        ?..(i)           
    • Differentiate with respect to x, \[2y.\frac{dy}{dx}=3p{{x}^{2}}\]           
    • Þ \[\frac{dy}{dx}=\frac{3p}{2}\left( \frac{{{x}^{2}}}{y} \right)\]           
    • \[\therefore {{\left| \frac{dy}{dx} \right|}_{2,3}}=\frac{3p}{2}\times \frac{4}{3}=2p\]           
    • For given line, slope of tangent \[=4\]           
    • \[\therefore 2p=4\] Þ \[p=2\]           
    • From equation (i),  \[9=2\times 8+q\] Þ\[q=-7\].


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