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JEE Main & Advanced Mathematics Conic Sections Question Bank Parabola

  • question_answer
    If the straight line \[x+y=1\] touches the parabola \[{{y}^{2}}-y+x=0\], then the co-ordinates of the point of contact are               [RPET 1991]

    A)            (1, 1)                                         

    B)            \[\left( \frac{1}{2},\ \frac{1}{2} \right)\]

    C)            (0, 1)                                         

    D)            (1, 0)

    Correct Answer: C

    Solution :

               m of tangent \[=-1\].            Also from equation of parabola, we get gradient at \[(h,k)\]as the slope of parabola                    \[=\frac{dy}{dx}=\frac{-1}{2y-1}=\frac{-1}{2k-1}\]                    Since line and parabola touch at \[(h,k)\]                    Þ \[\frac{-1}{2k-1}=-1\]Þ\[-2k+1=-1\]Þ\[k=1\]                    Putting this value in \[x+y=1\], we have \[h=0,\]so the point of contact is \[(0,\,1).\]


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