If the lines `(x-1)/2=(y+1)/3=(z-1)/4a n d(x-3)/1=(y-k)/2=z/1` intersect, then find the value of `kdot`

Explanation

The given lines are : `(x-1)/2 = (y+1)/3 = (z-1)/4 and (x-3)/1 = (y-k)/2 = z/4`
` (x-1)/2 = (y+1)/3 = (z-1)/4 = lambda` ...(i)
` :. P(2 lambda+1, 3 lambda-1,4lambda+1) `is any point on (i)
Also , let ` (x-3)/1 = (y-k)/2 = z/4 = mu ` ….(ii)
` :. Q (mu + 3, 2 mu + k, 4 mu)` is any point on (ii)
The two lines will intersect if
`2lambda+1 = mu +3`
` 3lambda -1 = 2 mu + k`
` 4 lambda + 1 = 4 mu ` ...(iii)
or ` mu = 2 lambda - 2` ...(iv)
or ` 3 lambda - 2 mu = k +1 ` ...(v)
Now, substituting (iv) in (iii),
` 4 lambda+1 = 4 (2 lambda-2)`
or ` 4 lambda+1 = 8 lambda - 8`
or ` 4 lambda = 9`
Substituting the value of `lambda ` in (iv) ,
` mu = 2 (9/4) - 2`
or ` mu = 9/2 - 2`
or ` mu = 5/2`
Substituting the value of `lambda and mu` in equation (v), we get
` 3 lambda - 2 mu = k+1`
or ` 3(9/4) - 2 (5/2) = k +1`
or ` 27/4 - 5 = k+1`
or ` 27/4 - 6 = k `
Thus , ` k = 3/4 `