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Question

If point C(¯¯c) divides the segment joining the point A(¯¯¯a) and B(¯¯b) internally in the ratio m:n, then prove that ¯¯c=m¯¯b+n¯¯¯am+n

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Solution

Let O be the origin.
Then AO=a and OB=b
Let P be a point on AB such that APPB=mn
n(AP)=m(PB)
n(OPOA)=m(OBOP)
(m+n)OP=mOB+nOA
(m+n)OP=mb+na
OP=c=mb+nam+n

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