If the lines given by `3x+2ky =2` and `2x+5y =1` are parallel, then the value of k is

Options

A. `-(5)/(4)`
B. `(2)/(5)`
C. `(15)/(4)`
D. `(3)/(2)`

Explanation

To find the value of \( k \) such that the lines given by the equations \( 3x + 2ky = 2 \) and \( 2x + 5y = 1 \) are parallel, we can follow these steps:

### Step 1: Write the equations in standard form
The first equation can be rewritten as:
\[ 3x + 2ky - 2 = 0 \]
This gives us:
- \( a_1 = 3 \)
- \( b_1 = 2k \)
- \( c_1 = -2 \)

The second equation can be rewritten as:
\[ 2x + 5y - 1 = 0 \]
This gives us:
- \( a_2 = 2 \)
- \( b_2 = 5 \)
- \( c_2 = -1 \)

### Step 2: Use the condition for parallel lines
For two lines to be parallel, the following condition must hold:
\[ \frac{a_1}{a_2} = \frac{b_1}{b_2} \quad \text{and} \quad \frac{a_1}{a_2} \neq \frac{c_1}{c_2} \]

### Step 3: Set up the ratio for \( a_1 \) and \( a_2 \)
Using the values we have:
\[ \frac{a_1}{a_2} = \frac{3}{2} \]

### Step 4: Set up the ratio for \( b_1 \) and \( b_2 \)
Using the values we have:
\[ \frac{b_1}{b_2} = \frac{2k}{5} \]

### Step 5: Equate the two ratios
Setting the two ratios equal gives us:
\[ \frac{3}{2} = \frac{2k}{5} \]

### Step 6: Cross-multiply to solve for \( k \)
Cross-multiplying gives:
\[ 3 \cdot 5 = 2 \cdot 2k \]
\[ 15 = 4k \]

### Step 7: Solve for \( k \)
Dividing both sides by 4:
\[ k = \frac{15}{4} \]

### Conclusion
The value of \( k \) such that the lines are parallel is:
\[ \boxed{\frac{15}{4}} \]