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. Then, PZ is

Options

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

Answer

A

Explanation

`(4((5a+4b)/(3))+(a+5b)/(5))/(4+1)=(21(a+b))/(25)=(21)/(25)PR`

PV of X is `(4(a+b)+a)/(5)=(5a+4b)/(5)`
PV of Y is `(4b+a+b)/(5)=(a+5b)/(5)`
Now, `PZ=mPR`
`PZ=m(a+b)`
Let Z divided YX in the ratio `lamda:1`
PV of `Z=(lamdaOX+OY)/(lamda+1)`
`therefore PZ=(((5a+4b)/(5))+(a+5b)/(5))/(lamda+1)=m(a+b)`
Comparing coefficients of a and b
`m=(5lamda+1)/(5(lamda+1))`
and `m=(4lamda+5)/(5(lamda+1))`
`therefore lamda=4`
`thereforePZ=(4((5a+4b)/(5))+(a+5b)/(5))/(4+1)`
`=(21(a+b))/(25)=(21)/(25)PR`.