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

  • question_answer
    The co-ordinates of the foot of perpendicular drawn from point \[P(1,\,0,\,3)\]to the join of points \[A(4,\,7,\,1)\]and \[B(3,\,5,\,3)\] is [RPET 2001]

    A) (5, 7, 1)

    B) \[\left( \frac{5}{3},\frac{7}{3},\frac{17}{3} \right)\]

    C) \[\left( \frac{2}{3},\frac{5}{3},\frac{7}{3} \right)\]

    D) \[\left( \frac{5}{3},\frac{2}{3},\frac{7}{3} \right)\]

    Correct Answer: B

    Solution :

    • Let D be the foot of perpendicular drawn from\[P(1,\,0,\,3)\] on the line AB joining (4, 7, 1) and (3, 5, 3).           
    • If D divides AB in ratio \[\lambda :1\] then \[D=\left( \frac{3\lambda +4}{\lambda +1},\,\frac{5\lambda +7}{\lambda +1},\,\frac{3\lambda +1}{\lambda +1} \right)\]                        .....(i)           
    • D.r-s of PD are \[2\lambda +3,\,5\lambda +7,\,-2\]           
    • D.r-s of AB are \[-1,\,-2,\,2\]           
    • \[\because \] \[PD\,\bot \,AB\]; \\[-(2\lambda +3)-2(5\lambda +7)-4=0\]Þ \[\lambda =\frac{-7}{4}\]           
    • Putting the value of \[\lambda \] in (i), we get the point \[D\left( \frac{5}{3},\,\frac{7}{3},\,\frac{17}{3} \right)\].


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