JEE Main & Advanced Mathematics Pair of Straight Lines Question Bank Critical Thinking

  • question_answer Area of the triangle formed by the lines \[{{y}^{2}}-9xy+18{{x}^{2}}=0\] and \[y=9\] is

    A)            \[\frac{27}{4}sq\]. units  

    B)            \[27sq.\]units

    C)            \[\frac{27}{2}sq.\] units  

    D)            None of these

    Correct Answer: A

    Solution :

               The lines represented by \[{{y}^{2}}-9xy+18{{x}^{2}}=0\] are \[6x-y=0\]and \[3x-y=0\]and third line is \[y=9\].            Therefore, coordinates of vertices of triangle are given by \[A(0,0)\];\[B\text{ }(3,9)\] and \[C\left( \frac{3}{2},9 \right)\].            Hence, area of DABC \[=\frac{1}{2}\left| \begin{matrix}    0 & 0 & 1  \\    3 & 9 & 1  \\    3/2 & 9 & 1  \\ \end{matrix} \right|=\frac{27}{4}sq.\,units.\]            Aliter: Applying the formula given in the theory part, the required area is            \[\frac{{{(-9)}^{2}}\sqrt{{{(9/2)}^{2}}-18}}{18\times 1+9\times 0\times 1+1\times 0}=\frac{81}{18}\sqrt{\frac{81}{4}-18}\]                                              \[=\frac{81}{18}\times \frac{3}{2}=\frac{27}{4}\,sq.\ units.\]


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