If the line `(x-4)/(1)=(y-2)/(1)=(z-q)/(p)` lies completely in the plane `2x-4y+z=7`, then the ordered pair (p, q) is

Options

A. (2, 7)
B. (7, 2)
C. (2, 4)
D. (1, 1)

Answer

To solve the problem, we need to determine the ordered pair \((p, q)\) such that the line given by

\[ \frac{x-4}{1} = \frac{y-2}{1} = \frac{z-q}{p} \]

lies completely in the plane defined by

\[ 2x - 4y + z = 7. \]

### Step-by-Step Solution:

1. **Parameterize the Line:**
From the given line equation, we can express \(x\), \(y\), and \(z\) in terms of a parameter \(\lambda\):
\[ x = 1\lambda + 4 = \lambda + 4, \]
\[ y = 1\lambda + 2 = \lambda + 2, \]
\[ z = p\lambda + q. \]

2. **Substitute into the Plane Equation:**
Substitute \(x\), \(y\), and \(z\) into the plane equation \(2x - 4y + z = 7\):
\[ 2(\lambda + 4) - 4(\lambda + 2) + (p\lambda + q) = 7. \]

3. **Simplify the Equation:**
Expanding the equation gives:
\[ 2\lambda + 8 - 4\lambda - 8 + p\lambda + q = 7. \]
Combine like terms:
\[ (2\lambda - 4\lambda + p\lambda) + (8 - 8 + q) = 7, \]
which simplifies to:
\[ (p - 2)\lambda + (q - 7) = 0. \]

4. **Set Coefficients to Zero:**
Since the equation must hold for all values of \(\lambda\), both coefficients must equal zero:
\[ p - 2 = 0, \]
\[ q - 7 = 0. \]

5. **Solve for \(p\) and \(q\):**
From \(p - 2 = 0\), we find:
\[ p = 2. \]
From \(q - 7 = 0\), we find:
\[ q = 7. \]

6. **Final Ordered Pair:**
Therefore, the ordered pair \((p, q)\) is:
\[ (2, 7). \]

### Summary:
The ordered pair \((p, q)\) is \((2, 7)\).