If the planes ` 2x- my +z=3 and 4x-y+2z` = 5 are parallel then m=

Options

A. `-2`
B. ` 2`
C. ` ( -1)/(2)`
D. ` (1)/(2) `

Answer

To determine the value of \( m \) for which the planes \( 2x - my + z = 3 \) and \( 4x - y + 2z = 5 \) are parallel, we can follow these steps:

### Step 1: Identify the coefficients of the planes
The general form of a plane is given by \( Ax + By + Cz = D \). For the first plane \( 2x - my + z = 3 \), the coefficients are:
- \( A_1 = 2 \)
- \( B_1 = -m \)
- \( C_1 = 1 \)

For the second plane \( 4x - y + 2z = 5 \), the coefficients are:
- \( A_2 = 4 \)
- \( B_2 = -1 \)
- \( C_2 = 2 \)

### Step 2: Set up the condition for parallel planes
For two planes to be parallel, the ratios of their corresponding coefficients must be equal. This gives us the following relationships:
\[ \frac{A_1}{A_2} = \frac{B_1}{B_2} = \frac{C_1}{C_2} \]

Substituting the coefficients we have:
\[ \frac{2}{4} = \frac{-m}{-1} = \frac{1}{2} \]

### Step 3: Solve the ratios
From the first ratio:
\[ \frac{2}{4} = \frac{1}{2} \]
This is true.

Now, from the second ratio:
\[ \frac{-m}{-1} = \frac{1}{2} \]
This simplifies to:
\[ m = \frac{1}{2} \]

### Step 4: Verify with the third ratio
Now, let's check the third ratio:
\[ \frac{1}{2} = \frac{1}{2} \]
This is also true.

### Conclusion
Thus, the value of \( m \) for which the planes are parallel is:
\[ \boxed{\frac{1}{2}} \]

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