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JAMIA MILLIA ISLAMIA Jamia Millia Islamia Solved Paper-2008

  • question_answer
        The tangent and the normal drawn to the curve\[y={{x}^{2}}-x+4\]at P(1, 4) tut the\[x-\]axis at A and B respectively. If the length of the subtangent drawn to the curve at P is equal to the length of the subnormal, then the area of the triangle PAB in sq unit is

    A)  4                                            

    B)  32

    C)  8                                            

    D)  16

    Correct Answer: D

    Solution :

                    Given equation of curve is \[y={{x}^{2}}-x+4\] Slope of tangent at\[P(1,4)\]is                 \[\left( \frac{dy}{dx} \right)=2x-1\] \[\Rightarrow \]               \[{{\left( \frac{dy}{dx} \right)}_{(1,4)}}=2-1=1\] \[\therefore \]Equation of tangent is \[y-4=1(x-1)\] \[\Rightarrow \]           \[y-x=3\]                            ...(i) and equation of normal at point\[P(1,4)\]is                 \[y-4=1(x-1)\]    \[\Rightarrow \]               \[x+y=5\]                                     ...(ii) Since, the tangent cuts x-axis at A \[\therefore \]Coordinates of A are\[(-3,0)\] and the normal cuts\[x-\]axis at B \[\therefore \]Coordinates of Bare (5, 0) \[\therefore \]Area of \[\Delta PAB\]                 \[=\frac{1}{2}\left| \left| \begin{matrix}    1 & 4 & 1  \\    -3 & 0 & 1  \\    5 & 0 & 1  \\ \end{matrix} \right| \right|\]                 \[=\frac{1}{2}|[-4(-3-5)]|=\frac{1}{2}|32|\] \[=16\]sq unit


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