If the line `(x-4)/(1)=(y-2)/(1)=(z-m)/(2)` lies in the plane `2x+ly+z=7`, then the value of `m+2l` is equal to

Options

A. 1
B. 2
C. `-1`
D. `-2`

Answer

To solve the problem, we need to determine the value of \( m + 2l \) given that the line

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

lies in the plane defined by the equation

\[ 2x + ly + z = 7. \]

### Step 1: Identify a point on the line

From the line equation, we can express the coordinates of a point on the line. Let \( t \) be the parameter:

\[ x = 4 + t, \quad y = 2 + t, \quad z = m + 2t. \]

### Step 2: Substitute the point into the plane equation

We need to substitute the coordinates of the point \( (x, y, z) \) into the plane equation \( 2x + ly + z = 7 \):

\[ 2(4 + t) + l(2 + t) + (m + 2t) = 7. \]

### Step 3: Expand and simplify

Expanding the equation gives:

\[ 8 + 2t + 2l + lt + m + 2t = 7. \]

Combining like terms results in:

\[ (2 + l + 2)t + (8 + m + 2l) = 7. \]

### Step 4: Set the coefficients equal to zero

For the line to lie in the plane for all \( t \), both the coefficient of \( t \) and the constant term must equal zero:

1. Coefficient of \( t \): \( 2 + l + 2 = 0 \)
2. Constant term: \( 8 + m + 2l = 7 \)

### Step 5: Solve the first equation for \( l \)

From the first equation:

\[ 2 + l + 2 = 0 \implies l + 4 = 0 \implies l = -4. \]

### Step 6: Substitute \( l \) into the second equation

Now substitute \( l = -4 \) into the second equation:

\[ 8 + m + 2(-4) = 7 \implies 8 + m - 8 = 7 \implies m = 7. \]

### Step 7: Calculate \( m + 2l \)

Now that we have \( m = 7 \) and \( l = -4 \), we can find \( m + 2l \):

\[ m + 2l = 7 + 2(-4) = 7 - 8 = -1. \]

Thus, the value of \( m + 2l \) is

\[ \boxed{-1}. \]