Which of the following pair of straight lines donot represent intersecting lines ?

Options

A. `y = (x)/(3) + (5)/(4) , y = (x)/(2) + (7)/(3)`
B. `3x - 4y = 0, x =0`
C. `4x + 3y = 1, y =0`
D. `2x + 3y = 7, 4x + 6y = 15`

Answer

To determine which pair of straight lines do not represent intersecting lines, we need to analyze the given options using the condition for non-intersecting lines. The condition is:

\[ \frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2} \]

Where \( a_1, b_1, c_1 \) are the coefficients from the first equation and \( a_2, b_2, c_2 \) are the coefficients from the second equation.

### Step-by-Step Solution:

**Step 1: Analyze Option A**
1. The equations are:
- \( y = x + 3 + \frac{y}{4} \)
- \( y = \frac{x}{2} + \frac{7}{3} \)

2. Rearranging the first equation:
- Multiply through by 4 to eliminate the fraction:
\[ 4y = 4x + 12 + y \implies 3y = 4x + 12 \implies -4x + 3y = 12 \]
- Coefficients: \( a_1 = -4, b_1 = 3, c_1 = 12 \)

3. Rearranging the second equation:
- Multiply through by 6 to eliminate the fraction:
\[ 6y = 3x + 14 \implies -3x + 6y = 14 \]
- Coefficients: \( a_2 = -3, b_2 = 6, c_2 = 14 \)

4. Calculate the ratios:
- \( \frac{a_1}{a_2} = \frac{-4}{-3} = \frac{4}{3} \)
- \( \frac{b_1}{b_2} = \frac{3}{6} = \frac{1}{2} \)
- \( \frac{c_1}{c_2} = \frac{12}{14} = \frac{6}{7} \)

5. Check the condition:
- \( \frac{4}{3} \neq \frac{1}{2} \) (True)
- Therefore, Option A does intersect.

**Step 2: Analyze Option B**
1. The equations are:
- \( 3x - 4y = 0 \)
- \( x + y = 0 \)

2. Coefficients:
- \( a_1 = 3, b_1 = -4, c_1 = 0 \)
- \( a_2 = 1, b_2 = 1, c_2 = 0 \)

3. Calculate the ratios:
- \( \frac{a_1}{a_2} = \frac{3}{1} = 3 \)
- \( \frac{b_1}{b_2} = \frac{-4}{1} = -4 \)
- \( \frac{c_1}{c_2} = \frac{0}{0} \) (undefined)

4. Check the condition:
- \( 3 \neq -4 \) (True)
- Therefore, Option B does intersect.

**Step 3: Analyze Option C**
1. The equations are:
- \( 4x + 3y = 1 \)
- \( y = 0 \)

2. Rearranging the second equation:
- \( 0x + 1y = 0 \)

3. Coefficients:
- \( a_1 = 4, b_1 = 3, c_1 = 1 \)
- \( a_2 = 0, b_2 = 1, c_2 = 0 \)

4. Calculate the ratios:
- \( \frac{a_1}{a_2} = \frac{4}{0} \) (undefined)
- \( \frac{b_1}{b_2} = \frac{3}{1} = 3 \)
- \( \frac{c_1}{c_2} = \frac{1}{0} \) (undefined)

5. Check the condition:
- Since \( \frac{a_1}{a_2} \) is undefined, we cannot conclude that they do not intersect.

**Step 4: Analyze Option D**
1. The equations are:
- \( 2x + 3y = 7 \)
- \( 4x + 6y = 15 \)

2. Coefficients:
- \( a_1 = 2, b_1 = 3, c_1 = 7 \)
- \( a_2 = 4, b_2 = 6, c_2 = 15 \)

3. Calculate the ratios:
- \( \frac{a_1}{a_2} = \frac{2}{4} = \frac{1}{2} \)
- \( \frac{b_1}{b_2} = \frac{3}{6} = \frac{1}{2} \)
- \( \frac{c_1}{c_2} = \frac{7}{15} \)

4. Check the condition:
- \( \frac{1}{2} = \frac{1}{2} \) and \( \frac{7}{15} \neq \frac{1}{2} \) (True)
- Therefore, Option D does not intersect.

### Conclusion:
The pair of straight lines that do not represent intersecting lines is **Option D**.