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JEE Main & Advanced Mathematics Differential Equations Question Bank Critical Thinking

  • question_answer
    The equation of family of curves for which the length of the normal is equal to the radius vector is

    A) \[{{y}^{2}}\pm {{x}^{2}}=k\]

    B) \[y\pm x=k\]

    C) \[{{y}^{2}}=kx\]                  

    D) None of these

    Correct Answer: A

    Solution :

    • Length of the normal \[=y\sqrt{1+{{\left( \frac{dy}{dx} \right)}^{2}}}\]        
    • It is given that \[y\sqrt{1+{{\left( \frac{dy}{dx} \right)}^{2}}}=\sqrt{{{x}^{2}}+{{y}^{2}}}\] (\[\because \]Radius vector \[=r=\sqrt{{{x}^{2}}+{{y}^{2}}}\])        
    • Þ \[{{y}^{2}}+{{y}^{2}}{{\left( \frac{dy}{dx} \right)}^{2}}={{x}^{2}}+{{y}^{2}}\] Þ \[{{y}^{2}}{{\left( \frac{dy}{dx} \right)}^{2}}={{x}^{2}}\]           
    • Þ \[ydy\pm xdx=0\] Þ \[{{y}^{2}}\pm {{x}^{2}}=k\].


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