Question

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 $PZ:ZR$

• A. $4:21$
• B. $3:4$
• C. $21:4$
• D. $4:3$

Answer 1

Answer: (C)

$21:4$

Let point $p$ be taken as origin and $\overrightarrow{q},\overrightarrow{s}$ are the position vectors of $Q$ and $S$ points respectively

$\Rightarrow \overrightarrow{PR}=\overrightarrow{q}+\overrightarrow{s}$

Position vector of $X=\frac{\overrightarrow{q}+4(\overrightarrow{q}+\overrightarrow{s})}{5}=\frac{5\overrightarrow{q}+4\overrightarrow{s}}{5}$

Position vector of $Y=\frac{4\overrightarrow{s}+\overrightarrow{q}+\overrightarrow{s}}{5}=\frac{\overrightarrow{q}+5\overrightarrow{s}}{5}$

Let $\frac{PZ}{ZR}=\frac{1}{\lambda }$ and $\frac{YZ}{ZX}=\mu$

position vector of $P=\frac{\overrightarrow{q}+\overrightarrow{s}}{\lambda +1}$

$\Rightarrow \frac{\overrightarrow{q}+\overrightarrow{s}}{\lambda +1}=\frac{\mu \left(\frac{5\overrightarrow{q}+4\overrightarrow{s}}{5}\right)+1\left(\frac{\overrightarrow{q}+5\overrightarrow{s}}{5}\right)}{\mu +1}$

$\Rightarrow \frac{\overrightarrow{q}+\overrightarrow{s}}{\lambda +1}=\frac{\mu (\overrightarrow{q}+\frac{4\overrightarrow{s}}{5})+(\frac{\overrightarrow{q}}{5}+\overrightarrow{s})}{\mu +1}$

$\Rightarrow \frac{\overrightarrow{q}+\overrightarrow{s}}{\lambda +1}=\frac{\overrightarrow{q}(\mu +\frac{1}{5})+\overrightarrow{s}\left(\frac{4\mu }{5}\right)}{\mu +1}$

By equating

$\therefore \frac{\overrightarrow{q}}{\lambda +1}=\frac{\overrightarrow{q}(\mu +\frac{1}{5})}{\mu +1}$

$\Rightarrow \frac{1}{\lambda +1}=\frac{(\mu +\frac{1}{5})}{\mu +1}$......$\left(1\right)$

and $\frac{\overrightarrow{s}}{\lambda +1}=\frac{\overrightarrow{s}(\frac{4\mu }{5}+1)}{\mu +1}$

$\Rightarrow \frac{1}{\lambda +1}=\frac{\frac{4\mu }{5}+1}{\mu +1}$....$\left(2\right)$

From $\left(1\right)$ and $\left(2\right)$ we get

$\frac{(\mu +\frac{1}{5})}{\mu +1}=\frac{\frac{4\mu }{5}+1}{\mu +1}$

$\Rightarrow \mu +\frac{1}{5}=\frac{4\mu }{5}+1$

$\Rightarrow \mu -\frac{4\mu }{5}=1-\frac{1}{5}$

$\frac{\mu }{5}=\frac{4}{5}\therefore \mu =4$

substitute $\mu =4$ in $\left(1\right)$ we get

$\frac{1}{\lambda +1}=\frac{4+\frac{1}{5}}{4+1}=\frac{21}{5\left(5\right)}=\frac{21}{25}$

$\Rightarrow \lambda +1=\frac{25}{21}$

$\Rightarrow \lambda =\frac{25}{21}-1=\frac{25-21}{21}=\frac{4}{21}$

$\Rightarrow \frac{PZ}{ZR}=\frac{21}{4}.$