11th Class Mathematics Conic Sections Question Bank Critical Thinking

  • question_answer Which one of the following curves cuts the parabola \[{{y}^{2}}=4ax\] at right angles      [IIT 1994]

    A)            \[{{x}^{2}}+{{y}^{2}}={{a}^{2}}\]      

    B)            \[y={{e}^{-x/2a}}\]

    C)            \[y=ax\]                                     

    D)            \[{{x}^{2}}=4ay\]

    Correct Answer: B

    Solution :

               \[{{y}^{2}}=4ax\]Þ\[2y{{\left( \frac{dy}{dx} \right)}_{1}}=4a\]Þ \[{{\left( \frac{dy}{dx} \right)}_{1}}=\frac{2a}{y}\]           ?..(i)                   Taking curve \[y={{e}^{-x/2a}}\]                   \[{{\left( \frac{dy}{dx} \right)}_{2}}={{e}^{-x/2a}}\left( -\frac{1}{2a} \right)\]\[=-\frac{y}{2a}\]                  .....(ii)                   Both curves cut orthogonally if,                    \[{{\left( \frac{dy}{dx} \right)}_{1}}{{\left( \frac{dy}{dx} \right)}_{2}}=-1\]Þ\[\left( -\frac{y}{2a} \right).\left( \frac{2a}{y} \right)=-1\].

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