The value of k for which the lines `x+2y=9` and `kx+4y= -5` are parallel, is

Options

A. `k=2`
B. `k=1`
C. `k= -1`
D. `k=3`

Answer

To find the value of \( k \) for which the lines \( x + 2y = 9 \) and \( kx + 4y = -5 \) are parallel, we can follow these steps:

### Step 1: Rewrite the equations in standard form
The first line can be rewritten as:
\[ x + 2y - 9 = 0 \]
This gives us the coefficients:
- \( a_1 = 1 \)
- \( b_1 = 2 \)
- \( c_1 = -9 \)

The second line can be rewritten as:
\[ kx + 4y + 5 = 0 \]
This gives us the coefficients:
- \( a_2 = k \)
- \( b_2 = 4 \)
- \( c_2 = 5 \)

### Step 2: Use the condition for parallel lines
For two lines to be parallel, the ratio of the coefficients of \( x \) and \( y \) must be equal. This can be expressed as:
\[ \frac{a_1}{a_2} = \frac{b_1}{b_2} \]
Substituting the values we found:
\[ \frac{1}{k} = \frac{2}{4} \]

### Step 3: Simplify the equation
The right side simplifies to:
\[ \frac{2}{4} = \frac{1}{2} \]
So we have:
\[ \frac{1}{k} = \frac{1}{2} \]

### Step 4: Solve for \( k \)
To find \( k \), we can cross-multiply:
\[ 1 \cdot 2 = 1 \cdot k \]
This simplifies to:
\[ k = 2 \]

### Conclusion
The value of \( k \) for which the lines are parallel is:
\[ \boxed{2} \]

---