Question

Consider the following equations of straight lines: 
\(\text{Line L1:}\) \( 2x - 3y = 5 \) 
\(\text{Line L2:}\) \( 3x + 2y = 8 \) 
\(\text{Line L3:}\) \( 4x - 6y = 5 \) 
\(\text{Line L4:}\) \( 6x - 9y = 6 \) 
Which one among the following is the correct statement?

• A. L1 is parallel to L2 and L1 is perpendicular to L3
• B. L2 is parallel to L4 and L2 is perpendicular to L1
• C. L3 is perpendicular to L4 and L3 is parallel to L2
• D. L4 is perpendicular to L2 and L4 is parallel to L3

Hint:

When determining the relationship between lines, use the slope formula. Lines are parallel if they have the same slope, and they are perpendicular if the product of their slopes is \( -1 \).

Answer 1

The Correct Option is D

We are given four straight lines. To determine the correct relationship between the lines, we need to calculate the slopes of the lines and then compare them.
Step 1: Find the slope of each line.
- The equation of line L1 is \( 2x - 3y = 5 \). Rewriting it in slope-intercept form \( y = mx + c \), we get: \[ 3y = 2x - 5 \quad \Rightarrow \quad y = \frac{2}{3}x - \frac{5}{3} \quad \text{(Slope of L1 is } \frac{2}{3} \text{)} \]
- The equation of line L2 is \( 3x + 2y = 8 \). Rewriting it: \[ 2y = -3x + 8 \quad \Rightarrow \quad y = -\frac{3}{2}x + 4 \quad \text{(Slope of L2 is } -\frac{3}{2} \text{)} \]
- The equation of line L3 is \( 4x - 6y = 5 \). Rewriting it: \[ 6y = 4x - 5 \quad \Rightarrow \quad y = \frac{2}{3}x - \frac{5}{6} \quad \text{(Slope of L3 is } \frac{2}{3} \text{)} \]
- The equation of line L4 is \( 6x - 9y = 6 \). Rewriting it: \[ 9y = 6x - 6 \quad \Rightarrow \quad y = \frac{2}{3}x - \frac{2}{3} \quad \text{(Slope of L4 is } \frac{2}{3} \text{)} \]

Step 2: Analyze the relationships.
- Lines L1, L3, and L4 have the same slope of \( \frac{2}{3} \), so they are parallel.
- Line L2 has a slope of \( -\frac{3}{2} \), which is different from the other lines, indicating that it is neither parallel nor perpendicular to L1, L3, or L4.
- Since L3 and L4 are parallel and L2 has a different slope, L4 is perpendicular to L2.
Thus, the correct statement is (D).