STATEMENT-1: The straight lines `2x+3y+5=0and 3x-2y+1=0` are perpendicular to each other.
STATEMENT-2: Two lines `y=m_(1)x+c_(1)and y=m_(2)x+c_(2)` where `m_(1),m_(2)in` R are perpendicular if `m_(1),m_(2)=-1.`
STATEMENT-2: Two lines `y=m_(1)xx+x_(1)and y=m_(2)x+c_(2)where m_(1),m_(2)inR` are peprendicular if `m_(1)m_(2)=-1.`

Options

A. Statement-1 is True, Statemetn-2 is True, Statemetn-2 is a correct explanation for Statement-1
B. Statement-1 is True, Statemetn-2 is True, Statement-2 is NOT a correct explanation for Statement-1
C. Statement-1 is True, Statement-2 is False
D. Statement-1 is False, Statement-2 is True

Answer

To determine whether the given statements about the lines are true, we will analyze each statement step by step.

### Step 1: Identify the slopes of the given lines

The equations of the lines are:
1. \( 2x + 3y + 5 = 0 \)
2. \( 3x - 2y + 1 = 0 \)

We need to convert these equations into the slope-intercept form \( y = mx + c \) to find their slopes.

**For the first line:**
\[ 3y = -2x - 5 \\ y = -\frac{2}{3}x - \frac{5}{3} \]
The slope \( m_1 \) of the first line is \( -\frac{2}{3} \).

**For the second line:**
\[ -2y = -3x - 1 \\ y = \frac{3}{2}x + \frac{1}{2} \]
The slope \( m_2 \) of the second line is \( \frac{3}{2} \).

### Step 2: Check if the lines are perpendicular

Two lines are perpendicular if the product of their slopes is equal to -1:
\[ m_1 \cdot m_2 = -1 \]
Substituting the values we found:
\[ -\frac{2}{3} \cdot \frac{3}{2} = -1 \]
Since this is true, the lines are indeed perpendicular.

### Step 3: Analyze the statements

**Statement 1:** The lines \( 2x + 3y + 5 = 0 \) and \( 3x - 2y + 1 = 0 \) are perpendicular to each other.
**Conclusion:** True, as we have shown that their slopes satisfy the condition for perpendicularity.

**Statement 2:** Two lines \( y = m_1x + c_1 \) and \( y = m_2x + c_2 \) are perpendicular if \( m_1 \cdot m_2 = -1 \).
**Conclusion:** True, as this is a standard condition for perpendicular lines.

### Final Conclusion

Both statements are true, and Statement 1 is correctly explained by Statement 2.