The equation of the line which passes through the point (1, 1, 1) and intersect the lines x−12=y−23=z−34 and x+21=y−32=z+14 is

[a] Any line passing through the point (1,1,1)is$x-aa=y-ab=z-ac\left(i\right)$This line intersects the line x−12=y−23$=z-34.$If a:b:c≠2:3:4 and ∣∣∣∣1−1a22−1b33−1c4∣∣∣∣=0$\Rightarrow a=2b+c=0???\left(ii\right)$Again, line (i) intersects line x−(−2)1=y−32$=z-\left(-1\right)4.$If a:b:c≠2:3:4and ∣∣∣∣−2−1a13−1b23−1c4∣∣∣∣=0$\Rightarrow 6a+5b-4c=0???.\left(iii\right)$From (ii) and (iii) by cross multiplication, we have$a8-5=b6+4=c5+12\text{or}3a=b10=c17$So, the required line is x−13=y−110=z−117.