JEE Main & Advanced Sample Paper JEE Main Sample Paper-34

  • question_answer
    In the xy-plane, three distinct lines  \[{{\ell }_{1}},{{\ell }_{2}},{{\ell }_{3}}\] are concurrent at \[M(\lambda ,0)\]. Also the lines\[{{\ell }_{1}},{{\ell }_{2}},{{\ell }_{3}}\], are normal's to the parabola \[{{y}^{2}}=6x\]at the points\[A\text{ }\left( {{x}_{1}},\text{ }{{y}_{1}} \right),B\left( {{x}_{2}},\text{ }{{y}_{2}} \right),C\left( {{x}_{3}},\text{ }{{y}_{3}} \right)\]respectively. Then

    A)  \[\lambda <-5\]

    B)  \[\lambda >3\]

    C)  \[-5>\lambda <-3\]

    D)  \[0>\lambda <3\]

    Correct Answer: B

    Solution :

    Any normal is \[y=mx-2am-a{{m}^{3}},\] where \[a=\frac{3}{2}\] \[\therefore \,\,(\lambda ,0)\,\,\Rightarrow 0=m\lambda -2am-a{{m}^{3}}\]             \[\Rightarrow m=0,{{m}^{2}}=\frac{\lambda }{a}-2>0\]             \[\Rightarrow \,\lambda \,>\,2a\,\Rightarrow \,\lambda \,>\,3\]

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