The distance between the point of concurrency of the two families of straight lines given by $\mathrm{x}+\left(5\mathrm{\lambda }+1\right)\mathrm{y}+1-3\mathrm{\lambda }=0$and  $\left(5\mathrm{\mu }+2\right)\mathrm{x}-3\mathrm{y}+3+6\mathrm{\mu }=0$ is    

Given   $\text{x\hspace{0.17em}+\hspace{0.17em}}\left(\text{5λ\hspace{0.17em}+\hspace{0.17em}1}\right)\text{\hspace{0.17em}y\hspace{0.17em}+\hspace{0.17em}1\hspace{0.17em}-\hspace{0.17em}3λ\hspace{0.17em}\hspace{0.17em}=\hspace{0.17em}0}$

$\left(\text{x\hspace{0.17em}+\hspace{0.17em}y\hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}1}\right)\text{\hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}λ\hspace{0.17em}}\left(\text{5y\hspace{0.17em}-\hspace{0.17em}3}\right)\text{\hspace{0.17em}=\hspace{0.17em}0\hspace{0.17em}\_\_\_\_\_\_}\left(\text{1}\right)$

${\text{L}}_{\text{1}}{\text{\hspace{0.17em}+\hspace{0.17em}λ\hspace{0.17em}L}}_{\text{2}}\text{\hspace{0.17em}=\hspace{0.17em}0}$

from (1) P.C  $\text{P\hspace{0.17em}=\hspace{0.17em}}\left(\frac{\text{-8}}{\text{5}}\text{,\hspace{0.17em}}\frac{\text{3}}{\text{5}}\right)$  and 

$\left(\text{5μ\hspace{0.17em}+\hspace{0.17em}2}\right)\text{\hspace{0.17em}x\hspace{0.17em}-\hspace{0.17em}3y\hspace{0.17em}+\hspace{0.17em}3\hspace{0.17em}+\hspace{0.17em}6μ\hspace{0.17em}=\hspace{0.17em}0}$

$\left(\text{2x\hspace{0.17em}-\hspace{0.17em}3y\hspace{0.17em}+\hspace{0.17em}3}\right)\text{\hspace{0.17em}+\hspace{0.17em}μ\hspace{0.17em}}\left(\text{5x\hspace{0.17em}+\hspace{0.17em}6}\right)\text{\hspace{0.17em}=\hspace{0.17em}0\hspace{0.17em}\hspace{0.17em}\_\_\_\_\_\_\_}\left(\text{2}\right)$

from (2) P.C    $\text{Q\hspace{0.17em}=\hspace{0.17em}}\left(\frac{\text{-6}}{\text{5}}\text{,\hspace{0.17em}\hspace{0.17em}}\frac{\text{1}}{\text{5}}\right)$

$\text{PQ\hspace{0.17em}=\hspace{0.17em}}\sqrt{{\left(\frac{\text{-8}}{\text{5}}\text{+\hspace{0.17em}}\frac{\text{6}}{\text{5}}\right)}^{\text{2}}\text{\hspace{0.17em}+\hspace{0.17em}}{\left(\frac{\text{3}}{\text{5}}\text{\hspace{0.17em}-}\frac{\text{1}}{\text{5}}\right)}^{\text{2}}}$

$\text{=\hspace{0.17em}}\frac{\text{2}\sqrt{\text{2}}}{\text{5}}$