Let the equation of a line which passes through the point (1,1,1) and intersects the lines $\frac{x-1}{1}=\frac{y-1}{1}=\frac{z-1}{1}\text{ and }\frac{x+2}{1}=\frac{y-3}{2}=\frac{z+1}{4}$ is $L_{1.}$ Which of the following statements is/are correct

Any line passing through the point (1,1,1) is $\frac{x-1}{a}=\frac{y-1}{b}=\frac{z-1}{c}$

This line intersects the line $\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}$

$\text{ If }a:b:c\ne 2:3:4\text{ and }\left|\begin{array}{ccc}1-1& 2-1& 3-1\\ a& b& c\\ 2& 3& 4\end{array}\right|=0$

$\Rightarrow a-2b+c=0\text{ Again, line (i) }$

intersects line $\frac{x-(-2)}{1}=\frac{y-3}{2}=\frac{z-(-1)}{4}$

$\text{ If }a:b:c\ne 1:2:4\text{ and }\left|\begin{array}{ccc}-2-1& 3-1& -1-1\\ a& b& c\\ 1& 2& 4\end{array}\right|=0$

$\Rightarrow 6a+5b-4c=0$

From (ii) and (iii) by cross multiplication, we have

$\frac{a}{8-5}=\frac{b}{6+4}=\frac{c}{5+12}\text{ or }\frac{a}{3}=\frac{b}{10}=\frac{c}{17}$

So, the required line is $\frac{x-1}{3}=\frac{y-1}{10}=\frac{z-1}{17}$