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

  • question_answer
    If the bisectors of the lines \[{{x}^{2}}-2pxy-{{y}^{2}}=0\] be \[{{x}^{2}}-2qxy-{{y}^{2}}=0,\] then              [MP PET 1993; DCE 1999; RPET 2003; AIEEE 2003; Kerala (Engg.) 2005]

    A)            \[pq+1=0\]                           

    B)            \[pq-1=0\]

    C)            \[p+q=0\]                              

    D)            \[p-q=0\]

    Correct Answer: A

    Solution :

               Bisector of the angle between the lines  \[{{x}^{2}}-2pxy-{{y}^{2}}=0\] is \[\frac{{{x}^{2}}-{{y}^{2}}}{xy}=\frac{1-(-1)}{-p}\]            \[\Rightarrow p{{x}^{2}}+2xy-p{{y}^{2}}=0\]            But it is represented by\[{{x}^{2}}-2qxy-{{y}^{2}}=0\].            Therefore\[\frac{p}{1}=\frac{2}{-2q}\Rightarrow pq=-1\].

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