Show that the lines
`(x-1)/(2)=(y-2)/(3)=(z-3)/(4) " and " (x-4)/(5)=(y-1)/(2)=z`
intersect each other . Find their point of intersection.

Explanation

The equations of the given lines are
`(x-1)/(2)=(y-2)/(3)=(z-3)/(4)=lambda`(say)
`(x-4)/(5)=(y-1)/(2)=(z-0)/(1)=,mu`
Any point on the line (i) is `P(2lambda+1,3lambda +2,4lambda +3)`
Any point on the line (ii) is `Q(5mu+4,2 mu+1,mu)`
If the lines (i) and (ii) intesect then P and Q must coincide for some particular values of `lambda " and " mu`
this gives
`2lambda +1= 5mu+ 4,3lambda+ 2=2mu +1 " and " 4lambda +3=mu`
`rArr {underset(4lambda -mu=-3)underset(3lambda -2mu =-1)(2lambda-5mu =3)`
On solving (i) and (ii) we get `lambda =-1 " and " mu =-1`
These values of `lambda " and " mu` also satify (iii)
Hence the given lines intersect .
putting `lambda=-1 " we get " P(-1,-1,-1)`
Note that putting `mu =-1 `we get Q(-1,-1,-1)
Hence thhe point of intersection of the given lines is (-1,-1,-1)