Which one of the following is correct is respect of the equations `(x -1)/(2) = (y - 2)/(3)` and `2x + 3y = 5` ?

Options

A. They represent two lines which are parallel
B. They represent two lines which are perpendicular
C. They represent two lines which are neither parallel nor perpendicular
D. The first equation does not represent a line

Answer

To determine the relationship between the equations \((x - 1)/(2) = (y - 2)/(3)\) and \(2x + 3y = 5\), we will follow these steps:

### Step 1: Rewrite the first equation in standard form
The first equation is given as:
\[ \frac{x - 1}{2} = \frac{y - 2}{3} \]
Cross-multiplying gives:
\[ 3(x - 1) = 2(y - 2) \]
Expanding both sides:
\[ 3x - 3 = 2y - 4 \]
Rearranging it to standard form:
\[ 3x - 2y = -1 \quad \text{(Equation 1)} \]

### Step 2: Rewrite the second equation in standard form
The second equation is:
\[ 2x + 3y = 5 \quad \text{(Equation 2)} \]

### Step 3: Find the slopes of both lines
To find the slopes, we will convert both equations into the slope-intercept form \(y = mx + b\).

**For Equation 1:**
\[ 3x - 2y = -1 \]
Rearranging gives:
\[ -2y = -3x - 1 \implies y = \frac{3}{2}x + \frac{1}{2} \]
Thus, the slope \(m_1\) is:
\[ m_1 = \frac{3}{2} \]

**For Equation 2:**
\[ 2x + 3y = 5 \]
Rearranging gives:
\[ 3y = -2x + 5 \implies y = -\frac{2}{3}x + \frac{5}{3} \]
Thus, the slope \(m_2\) is:
\[ m_2 = -\frac{2}{3} \]

### Step 4: Determine if the lines are perpendicular
Two lines are perpendicular if the product of their slopes is \(-1\):
\[ m_1 \cdot m_2 = \left(\frac{3}{2}\right) \cdot \left(-\frac{2}{3}\right) = -1 \]

Since the product of the slopes is \(-1\), the lines represented by the equations are indeed perpendicular.

### Conclusion
The correct answer is that the two lines represented by the equations are perpendicular to each other.

---