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JEE Main & Advanced Mathematics Three Dimensional Geometry Question Bank Critical Thinking

  • question_answer
    If the lines \[\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-1}{4}\] and \[\frac{x-3}{1}=\frac{y-k}{1}=\frac{z}{1}\] intersect, then k = [IIT Screening 2004]

    A) \[\frac{2}{9}\]

    B) \[\frac{9}{2}\]

    C) 0

    D) None of these

    Correct Answer: B

    Solution :

    • Any point on \[\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-1}{4}\]\[=\lambda \] is,                   
    • \[(2\lambda +1,\,3\lambda -1,\,4\lambda +1);\,\,\lambda \in R\]                   
    • Any point on \[\frac{x-3}{1}=\frac{y-k}{2}=\frac{z}{1}\]\[=\mu \] is,                   
    • \[(\mu +3,\,2\mu +k,\,\mu );\,\mu \in R\]                   
    • The given lines intersect if and only if the system of equations (in \[\lambda \,\] and \[\mu \])                                
    • \[2\lambda +1=\mu +3\]                              .....(i)                                
    • \[3\lambda -1=2\mu +k\]                             .....(ii)                                
    • \[A+2B+3C=0\]                                           .....(iii)                   
    • has a unique solution.                   
    • Solving (i) and (iii), we get \[\lambda =\frac{-3}{2},\,\mu =-5\]           
    • From (ii), we get \[\frac{-9}{2}-1=-10+k\Rightarrow k=\frac{9}{2}\].             


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