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NDA2018MathematicsThree Dimensional GeometryActual

Let the coordinates of the points A , B , C be (1,8,4) , (0,-11,4) and (2,-3,1) respectively. What are the coordinates of the point D which is the foot of the perpendicular from A on BC ?

Options

  1. A(3,4,-2)
  2. B(4,-2,5)
  3. C(4,5,-2)
  4. D(2,4,5)

Correct answer

C. (4,5,-2)

Step-by-step solution

aligned & A (1,8,4), B (0,-11,4), C (2,-3,1) & Let D =( , , ) aligned Direction ratios of BC = ( x ₂- x ₁, y ₂- y ₁, z ₂- z ₁ ) Let, ( a , b , c )=(2,8,-3) Direction ratios of A D= ( x ₂- x ₁, y ₂- y ₁, z ₂- z ₁ ) Let (a^ , b^ , c^ )=( -1, -8, -4) Since, A D is perpendicular to B C , aligned & aa ^ + bb ^ + cc ^ =0 & 2( -1)+8( -8)-3( -4)=0 & 2 -2+8 -64-3 +12=0 & 2 +8 -3 -54=0...(1) aligned On substituting the options, we find option (c) is correct. when ( , , )=(4,5,-2) aligned & (1) 2(4)+8(5)-3(-2)-54=0 & 8+40+6-5

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