JEE Main & Advanced Mathematics Three Dimensional Geometry Question Bank Plane

  • question_answer
    The points \[A(-1,3,0)\], \[B\,(2,\,2,\,1)\] and \[C\,(1,\,1,\,3)\] determine a plane. The distance from the plane to the point \[D(5,\,7,8)\] is                                                               [AMU 2001]

    A)            \[\sqrt{66}\]

    B)            \[\sqrt{71}\]

    C)            \[\sqrt{73}\]

    D)            \[\sqrt{76}\]

    Correct Answer: A

    Solution :

               Equation of plane passing through \[(-1,\,3,\,0)\] is \[A(x+1)+B(y-3)+C(z-0)=0\]                     ......(i)            Also, plane (i) is passing through the points \[(2,\,2,\,1)\] and (1, 1, 3).  So,       \[3A-B+C=0\]                     .....(ii)                                 \[2A-2B+3C=0\]                .....(iii)            Solving (ii) and (iii), \[\frac{A}{-3+2}=\frac{B}{2-9}=\frac{C}{-6+2}\]            \ \[A:B:C=-1:-7:-4\] or \[A:B:C=1:7:4\]            From (i), \[1(x+1)+7(y-3)+4(z)=0\]            or \[x+7y+4z-20=0\]            \[\therefore \]Distance from the plane to the point (5, 7, 8) is,                  \[\frac{1\times 5+7\times 7+4\times 8-20}{\sqrt{{{1}^{2}}+{{7}^{2}}+{{4}^{2}}}}=\frac{5+49+32-20}{\sqrt{66}}=\frac{66}{\sqrt{66}}=\sqrt{66}\].

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