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Question

Given that P(3, 2, 4); Q(5, 46); R(9, 8, 10) are collinear, find the ratio in which Q divides [PR].

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Solution

P(3,2,4),Q(5,4,6),R(9,8,10) are collinear
Q must divide line segment PR in some ratio externally and internally
(x,y,z)=(mx2+nx1m+n,my2+ny1m+n)
Q(5,4,6)=(m(9)+n(3)m+n,m(8)+n(2)m+n,m(10)+n(4)m+n)
m=k,n=1
(5,4,6)=(9k+3k+1,8k+2k+1,10k4k+1)
5=9k+3k+1
5(k+1)=9k+3
9k5k=53
4k=2 k=12
i.e., m:n=k:1
=12:1
=1:2
Point Q divides PR in the ratio 1:2

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