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Question

ABCD is a tetrahedron where A(2,0,0), B(0,4,0) and CD=14. The edge CD lies on the line x11=y22=z33. If locus of centroid of tetrahedron is x321=yy1a=zz1b, then

A
a+b=5
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B
y1+z1=6
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C
y1z1=1
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D
a+b+y1=8
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Solution

The correct option is D a+b+y1=8
Given : A(2,0,0),B(0,4,0) and CD lies on the line x11=y22=z33
Let C(λ1+1,2λ1+2,3λ1+3)
and D(λ2+1,2λ2+2,3λ2+3)
CD=14
(λ1λ2)2+4(λ1λ2)2+9(λ1λ2)2=14
λ1λ2=±1
Taking negative sign, we get
λ2=1+λ1
C(λ1+1,2λ1+2,3λ1+3) and
D(λ1+2,2λ1+4,3λ1+6)

Let centroid of tetrahedron be G(α,β,γ)
α=2+0+λ1+1+λ1+24
4α=5+2λ1
Similarly, 4β=10+4λ1
and 4γ=9+6λ1
Thus, locus of the centroid of tetrahedron is 4x52=4y104=4z96
x5412=y521=z9432
x541212=y52112=z943212
x321=y32=z33
y1=3=z1,a=2 and b=3

Now, a+b=2+3=5
y1+z1=3+3=6
y1z1=33=0
and a+b+y1=2+3+3=8

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