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JEE Main & Advanced Mathematics Pair of Straight Lines Question Bank Angle between the pair of straight lines, Condition for parallel and perpendicular lines

  • question_answer
    If the angle between the pair of straight lines represented by the equation \[{{x}^{2}}-3xy+\lambda {{y}^{2}}+3x-5y+2=0\] is \[{{\tan }^{-1}}\left( \frac{1}{3} \right)\], where \['\lambda \,'\]is a non negative real number. Then\[\lambda \]is [Orissa JEE 2002]

    A)            2    

    B)            0

    C)            3    

    D)            1

    Correct Answer: A

    Solution :

               Given that \[\theta ={{\tan }^{-1}}\left( \frac{1}{3} \right)\] Þ \[\tan \theta =\frac{1}{3}\]            Now, since \[\tan \theta =\frac{2\sqrt{{{h}^{2}}-ab}}{a+b}\]Þ \[\frac{1}{3}\,=\frac{2\sqrt{{{\left( \frac{-3}{2} \right)}^{2}}-\lambda }}{\lambda +1}\]            Þ \[{{(\lambda +1)}^{2}}=9(9-4\lambda )\] Þ \[{{\lambda }^{2}}+38\lambda -80=0\]            Þ \[\lambda =\frac{-38\pm \sqrt{{{(38)}^{2}}+320}}{2}\]Þ \[\lambda =\frac{-38\pm 42}{2}\]Þ \[\lambda =2\].


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