Find the equation of a line which passes through the point `(1,1,1)` and intersects the lines `(x-1)/2=(y-2)/3=(z-3)/4a n d(x+2)/1=(y-3)/2=(z+1)/4dot`

Explanation

Any line passing through the point (1, 1, 1) is `(x-1)/(a)=(y-1)/(b)=(z-1)/(c)" "`(i)
This line intersects the line `(x-1)/(2)=(y-2)/(3)=(z-3)/(4)`.
If `a:b:c ne 2:3:4 and |{:(1-1,,2-1,,3-1),(a,,b,,c),(2,,3,,4):}|=0`
`rArr" "a-2b+c=0`
Again, line (i) intersects line `(x-(-2))/(1)=(y-3)/(2)=(z-(-1))/(4)" "`(ii)
`a:b:c ne 1:2:4 and |{:(-2-1,,3-1,,-1-1),(a,,b,,c),(1,,2,,4):}|=0`
`rArr" "6a+5b-4c=0" "`(iii)
From (ii) and (iii) by cross multiplication, we have `(a)/(8i-5)= (b)/(6+4)=(c)/(5+12)`
or `" "(a)/(3)=(b)/(10)=(c)/(17)`
So, the required line is `(x-1)/(3)=(y-1)/(10)=(z-1)/(17)`