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Question

Show that the points O(0,0),A(2,3,3),B(2,3,3) are collinear. Find the ratio in which each point divides the segment joining the other two.

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Solution

Let's find area of OAB which is given by,
Δ=12|OA||OB|sinθ
where θ is angle between OA and OB
Now, cosθ=OA.OAOAOB
=4994+9+94+9+9=2222
cosθ=1 θ=180o
Δ=0 as sin180=0
O,A,B are collinear
Now, let A divides OB in k:1
2=2k+0k+1 2k+2=2k 4k=2 k=12
A divides OB in 1:2 externally
Now for B
2=2k+0k+1 2k2=2k 4k=2
k=12
B divides OA in 1:2 externally for O
O=2k+2k+1 2k=2 k=1
O divides AB in 1:1 internally

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