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.

Options

A. `PX=(1)/(5)PR`
B. `PX=(3)/(5)PR`
C. `PX=(2)/(5)PR`
D. none of these

Answer

B

Explanation

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`