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CET Karnataka Engineering CET - Karnataka Engineering Solved Paper-2006

  • question_answer
    The tangent and the normal drawn to the curve \[y={{x}^{2}}-x+4\] at P(1, 4) cut the x-axis at A and B respectively. If the length of the sub tangent 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                 \[\left( \frac{dy}{dx} \right)=2x-1\] \[\Rightarrow \]               s\[{{\left( \frac{dy}{dx} \right)}_{(1,4)}}=2-1=1\] \[\therefore \] 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 \[\therefore \]                  Co-ordinates of A are \[(-3,0)\] and the normal cuts x-axis at B \[\therefore \]  Co-ordinates of Bare \[(5,0)\] Area of \[\Delta PAB\]                 \[=\frac{1}{2}\left\| \begin{matrix}    1 & 4 & 1  \\    -3 & 0 & 1  \\    5 & 0 & 1  \\ \end{matrix} \right\|\]                 \[=\frac{1}{2}|[-4(-3-5)]|=\frac{1}{2}|32|\]                 \[=16sq\] unit


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