If the distance between the plane 23x-10y-2z+48 =0 and the plane containing the lines `(x+1)/2 =(y-3)/4= (z+1)/(3)` and `(x+3)/(2)=(y+2)/6 = (z-1)/(lambda)(lambda in R)` is equal to `k/sqrt(633)` , then k is equal to __________.

Answer

To solve the problem, we need to follow these steps:

### Step 1: Identify the lines and their parametric equations
The two lines given are:
1. \((x+1)/2 = (y-3)/4 = (z+1)/3\)
2. \((x+3)/2 = (y+2)/6 = (z-1)/\lambda\)

We can express these lines in parametric form:
- For the first line, let \( t \) be the parameter:
\[ x = 2t - 1, \quad y = 4t + 3, \quad z = 3t - 1 \]

- For the second line, let \( p \) be the parameter:
\[ x = 2p - 3, \quad y = 6p - 2, \quad z = \lambda p + 1 \]