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In a triangle P Q R ,Sa n dT are points ...

In a triangle `P Q R ,Sa n dT` are points on `Q Ra n dP R ,` respectively, such that `Q S=3S Ra n dP T=4T Rdot` Let `M` be the point of intersection of `P Sa n dQ Tdot` Determine the ratio `Q M : M T` using the vector method .

Text Solution

Verified by Experts

The correct Answer is:
`15 : 4`

Let `QM : MT= lamda : 1 and PM : MS= mu : 1`
and `vec(QP)= veca, vec(QP)= vecb`
`rArr " "vec(QT)= (4vecb+veca)/(5)`
and `" "vec(QM) = (lamda )/(lamda +1) ((4vecb+ veca)/(5))" "` (i)
`" "vec(QS) =(3)/(4) vecb, vec(QM) = (mu((3)/(4) vecb)+ veca)/(mu+1)" "` (ii)
From (i) and (ii), we have
`" "(1)/(mu+1) =(lamda )/(5(lamda + 1)) and (4lamda)/(5(lamda +1))= (3mu)/(4(mu+1))`
`rArr lamda = 15//4 and mu= 16//3 `
`therefore " "QM: MT= 15: 4`
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