Find the values of a so that the following lines are skew.
`(x-1)/(2) = (y-2)/(3) = (z-a)/(4), (x-4)/(5) = (y-1)/(2) = z`.

Answer

To find the values of \( a \) such that the given lines are skew, we start by rewriting the equations of the lines in parametric form.

### Step 1: Write the equations in parametric form

For the first line:
\[ \frac{x-1}{2} = \frac{y-2}{3} = \frac{z-a}{4} = \lambda \]
From this, we can express \( x, y, z \) in terms of \( \lambda \):
\[ x = 2\lambda + 1 \]
\[ y = 3\lambda + 2 \]
\[ z = 4\lambda + a \]

For the second line:
\[ \frac{x-4}{5} = \frac{y-1}{2} = z = \mu \]
From this, we can express \( x, y, z \) in terms of \( \mu \):
\[ x = 5\mu + 4 \]
\[ y = 2\mu + 1 \]
\[ z = \mu \]

### Step 2: Set the equations equal to find conditions for skew lines

For the lines to be skew, they must not intersect and must not be parallel. We can set the parametric equations equal to each other:
1. \( 2\lambda + 1 = 5\mu + 4 \)
2. \( 3\lambda + 2 = 2\mu + 1 \)
3. \( 4\lambda + a = \mu \)

### Step 3: Solve the equations

From equation 1:
\[ 2\lambda + 1 = 5\mu + 4 \implies 2\lambda - 5\mu = 3 \tag{1} \]

From equation 2:
\[ 3\lambda + 2 = 2\mu + 1 \implies 3\lambda - 2\mu = -1 \tag{2} \]

Now we have a system of linear equations (1) and (2). We can solve for \( \lambda \) and \( \mu \).

### Step 4: Solve the system of equations

Multiply equation (1) by 2:
\[ 4\lambda - 10\mu = 6 \tag{3} \]

Now multiply equation (2) by 5:
\[ 15\lambda - 10\mu = -5 \tag{4} \]

Now, subtract equation (3) from equation (4):
\[ (15\lambda - 10\mu) - (4\lambda - 10\mu) = -5 - 6 \]
\[ 11\lambda = -11 \implies \lambda = -1 \]

Substituting \( \lambda = -1 \) back into equation (1):
\[ 2(-1) - 5\mu = 3 \implies -2 - 5\mu = 3 \implies -5\mu = 5 \implies \mu = -1 \]

### Step 5: Substitute \( \lambda \) and \( \mu \) into the third equation

Now substitute \( \lambda = -1 \) and \( \mu = -1 \) into the third equation:
\[ 4(-1) + a = -1 \implies -4 + a = -1 \implies a = 3 \]

### Conclusion

Thus, the value of \( a \) such that the lines are skew is:
\[ \boxed{3} \]