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Question

A tetrahedron is a three dimensional figure bounded by four non-coplanar triangular planes. So, a tetrahedron has four non-coplanar points as its vertices.
Let a tetrahedron has four points A, B, C, D whose co-ordinates are (xi,yi,zi), i=1, 2, 3, 4 respectively in a rectangular three dimensional space. The co-ordinates of its centroid are given by 4i=1xi4,4i=1yi4,4i=1zi4. The circumcenter of the tetrahedron is the center of a sphere passing through its vertices. So this point is equidistant from each of vertices of tetrahedron.
Let tetrahedron has three of its vertices represented by the points A (6, - 5, -1), B(- 4, 1, 3) & C(2, -4, 18) & its centroid lies at the point (1, -2, 5).
On the basis of above information answer the following questions.The co-ordinate of centre of the sphere circumscribe the tetrahedron is

A
(87,457,8)
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B
(87,457,8)
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C
(87,457,8)
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D
None of these
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Solution

The correct option is C (87,457,8)
Let P(a, b, c) be the centre of the sphere circumscribing the tetrahedron. Let r be the radius of the sphere.
r2=(a0)2+(b0)2+(c0)2=a2+b2+c2 (i)
(PA)2=(a6)2+(b+5)2+(c+1)2=a2+b2+c212a+10b+2c+62 (ii)
(PB)2=(a+4)2+(b1)2+(c3)2=a2+b2+c2+8a2b6c+26 (iii)
(PC)2=(a2)2+(b+4)2+(c18)2=a2+b2+c24a+8b36c+344 (iv)
From (i) & (iv) a2b+9c=86
From (i) & (iii) 4ab3c=13
From (i) & (ii) 6a5bc=31
a∣ ∣298613135131∣ ∣=b∣ ∣198643136131∣ ∣=c∣ ∣1286413651∣ ∣
=a128=b720=c896=1112
a=128112, b=720112, c=896112
(a,b,c)=(87,457,8)

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