Show that the lines `(x+1)/(3)=(y+3)/(5) =(z+5)/(7) " and " (x-2)/(1)=(y-4)/(3)=(z-6)/(5)`
intersect. Also find their point of intersection.

Explanation

The given lines are
`(x-1)/(3)=(y+3)/(5)=(z+5)/(7)=lambda `(say)
and `(x-2)/(1)= (y-4)/(3)=(z-6)/(5)=mu` (say)
The general point on (1) is P `(3lambda -1 ,5lambda -3,7lambda -5)`
The general point on (2) is Q `(mu +2 ,3mu +4, 5mu +6)`
The given lines will intersect only when they have a common point. This happens when P and Q coincide for some particular values of `lambda " and " mu`
So , the given lines will intersect only when
`3lambda-1=mu +2,5lambda-3=3mu +4 "and " 7lambda -5=5mu+6`
`rArr 3lambda -mu =3....(i) ,5lambda -3mu,=7 .....(ii) "and " 7lambda -5mu =11`
On solving (i) and (ii) we get `lambda =(1)/(2) " and " mu =(-3)/(2)`
Clearly these values of `lambda " and " mu ` also satisfy (iii)
Hence the given lines intersect .
Putting `lambda=(1)/(2) " in P or " mu =(-3)/(2) ` in Q we get the point of intersection of the given lines as `((1)/(2),(-1)/(2),(-3)/(2))`