Points X and Y are taken on the sides QR and RS, respectively of a parallelogram PQRS, so that `QX=4XR` and `RY=4YS` The line XY cuts the line PR at Z Find the ratio PZ: ZR

Options

A. `21/25 vec(PR)`
B. `16/25 vec(PR)`
C. `17/25vec(PR)`
D. None of these

Answer

A

Explanation

`(4(5veca+vecb))/(5)+(veca+5vecb)/(5) = (21(veca+vecb))/(25) =21/25 vec(PR)`

P.V. of X is `(4(veca+vecb))/(5) =(5veca+4vecb)/(5)`
P.V.of Y is `(4vecb+veca+vecb)/(5) =(veca+5vecb)/(5)`
Now, `vec(PZ) = mvec(PR)`
`vec(PZ) = m(veca+vecb)`
Let Z divides YX in the ratio `lambda:1`
P.V. of `Z=(lambdavec(OX)+vec(OY))/(lambda+1)`
`therefore (vec(PZ)) = (lambda(5veca+vecb))/(5) + ((veca+5vecb)/(5))/(lambda+1) =m(veca+vecb)`
Comparing coefficients of `veca` and `vecb`
`m=(5lambda+1)/(5(lambda+1))` and `m=(4lambda+5)/(5(lambda+1))`
`therefore lambda=4`
`therefore lambda(PZ) = ((4(5veca+4vecb))/(5) + (veca+5vecb)/(5))/(4+1)`
`=(21(veca+vecb))/(25) = 21/25 vec(PR)`