Let OABCD be a pentagon in which the sides OA and CB are parallel and the sides OD and AB are parallel. Also OA:CB=2:1 and OD:AB=1:3

Then the ratio $\frac{OX}{XC}$ is___

Let the position vectors of A,B,C and D be $\bar{a},\bar{b},\bar{c}$ and $\bar{d}$ respectively.
Then OA:CB = 2 :1

$\Rightarrow \overrightarrow{a}=2(\overrightarrow{b}-\overrightarrow{c})\dots .\left(1\right)$

And OD: AB =1: 3

$$\begin{gathered} \stackrel{-}{3OD}=\stackrel{-}{AB} \\ 3\overrightarrow{d}=(\overrightarrow{b}-\overrightarrow{a})=\overrightarrow{b}-2(\overrightarrow{b}-\overrightarrow{c}) \end{gathered}$$

Using (1)

$\Rightarrow 3\overrightarrow{d}=-\overrightarrow{b}+2\overrightarrow{c}$

Let $OX:XC=\lambda :1$ and $AX:XD=\mu :1$

Now, x divides OC in the ratio $\lambda :1$ and , Therefore, P.V of $\mathrm{x}=x=\frac{\lambda \overrightarrow{c}}{\lambda +1}$ …..(3)
X also divides AD in the ratio $\mu :1$ P.V of $x=\frac{\mu \overrightarrow{d}+\overrightarrow{a}}{\mu +1}$

$\frac{\lambda \overrightarrow{c}}{\lambda +1}=\frac{\mu \overrightarrow{d}+\overrightarrow{a}}{\mu +1}$

$\Rightarrow \left(\frac{\lambda \overrightarrow{c}}{\lambda +1}\right)\overrightarrow{c}=\left(\frac{\mu }{\mu +1}\right)\overrightarrow{d}+\left(\frac{1}{\mu +1}\right)\overrightarrow{a}$

$\Rightarrow \left(\frac{\lambda \overrightarrow{c}}{\lambda +1}\right)\overrightarrow{c}=\left(\frac{\mu }{\mu +1}\right)\left(\frac{-\overrightarrow{b}+2\overrightarrow{c}}{3}\right)+\left(\frac{1}{\mu +1}\right)2(\overrightarrow{b}-\overrightarrow{c})$

Using (1) and (4)

$\Rightarrow \left(\frac{\lambda \overrightarrow{c}}{\lambda +1}\right)\overrightarrow{c}=\left(\frac{6-\mu }{3(\mu +1)}\right)\overrightarrow{b}+\left(\frac{2\mu }{3(\mu +1)}-\frac{2}{\mu +1}\right)\overrightarrow{c}$$\Rightarrow \left(\frac{6-\mu }{3(\mu +1)}\right)\overrightarrow{b}+\left(\frac{2\mu -6}{3(\mu +1)}-\frac{\lambda }{\mu +1}\right)\overrightarrow{c}=\overrightarrow{0}$

[As $\overrightarrow{b}$ and $\overrightarrow{c}$ are non - coplanar ]

$\Rightarrow \mu =6,\mathrm{ }\lambda =\frac{2}{3}$

Hence, OX : XC = 2 : 5