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In a quadrilateral P Q R S , vec P Q= ve...

In a quadrilateral `P Q R S , vec P Q= vec a , vec Q R = vec b , vec S P= vec a- vec b ,M` is the midpoint of ` vec Q Ra n dX` is a point on `S M` such that `S X=4/5S Mdot` Prove that `P ,Xa n dR` are collinear.

A

`PX=(1)/(5)PR`

B

`PX=(3)/(5)PR`

C

`PX=(2)/(5)PR`

D

none of these

Text Solution

Verified by Experts

The correct Answer is:
B
If we take point P as the origin, the position vectors of Q and S are a and b-a respectively.
In `DeltaPQR`, we have

`PR=PQ+QR impliesPR=a+b`
`therefore`Position vector of R=a+b
`impliesPV` or `M=(a+(a+b))/(2)=(a+(1)/(2)b)`
Now, `SX=(4)/(5)SM`
`impliesXM=SM-SX=SM-(4)/(5)SM=(1)/(5)SM`
`thereforeSX:XM=4:1`
`impliesPV` of `X=(4(a+(1)/(2)b)+1(b-a))/(4+1)`
`=(3a+2b)/(5)impliesPX=(3)/(5)(a+b)`
`impliesPX=(3)/(5)PR`
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