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

  • question_answer The area (in square units) of the quadrilateral formed by the two pairs of lines\[{{l}^{2}}{{x}^{2}}-{{m}^{2}}{{y}^{2}}-n(lx+my)=0\]and  \[{{l}^{2}}{{x}^{2}}-{{m}^{2}}{{y}^{2}}+n(lx-my)=0\] is                [EAMCET 2003]

    A)            \[\frac{{{n}^{2}}}{2|lm|}\]    

    B)            \[\frac{{{n}^{2}}}{|lm|}\]

    C)            \[\frac{n}{2|lm|}\]            

    D)            \[\frac{{{n}^{2}}}{4|lm|}\]

    Correct Answer: A

    Solution :

               Given lines are (on factorising)            \[lx+my=0,\,lx+my+n=0\]            \[lx-my=0,\,lx+my-n=0\]            Area =\[\left| \frac{({{c}_{1}}-{{d}_{1}})\,({{c}_{2}}-{{d}_{2}})}{({{a}_{1}}{{b}_{2}}-{{a}_{2}}{{b}_{1}})} \right|=\left| \frac{(0-n)\,(0+n)}{(-lm-lm)} \right|=\frac{{{n}^{2}}}{2|lm|}\].

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