If the straight lines `x+2y=3, 2x+3y=5 and k^(2)x+ky=-1` represent a triangle which is right - angled, then the value of k are `k_(1) and k_(2)`. The value of `|(k_(1)+k_(2))/(k_(1)-k_(2))|` is

Answer

To solve the problem step by step, we need to determine the values of \( k \) such that the lines \( x + 2y = 3 \), \( 2x + 3y = 5 \), and \( k^2 x + ky = -1 \) form a right-angled triangle.

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

1. **First line:** \( x + 2y = 3 \)
- Rearranging gives \( 2y = -x + 3 \) or \( y = -\frac{1}{2}x + \frac{3}{2} \)
- Slope \( m_1 = -\frac{1}{2} \)

2. **Second line:** \( 2x + 3y = 5 \)
- Rearranging gives \( 3y = -2x + 5 \) or \( y = -\frac{2}{3}x + \frac{5}{3} \)
- Slope \( m_2 = -\frac{2}{3} \)

3. **Third line:** \( k^2 x + ky = -1 \)
- Rearranging gives \( ky = -k^2 x - 1 \) or \( y = -\frac{k^2}{k}x - \frac{1}{k} \)
- Slope \( m_3 = -k \)

### Step 2: Conditions for a right-angled triangle

For the triangle to be right-angled, the product of the slopes of any two lines must equal -1. We will check the combinations:

1. **Between the first and second lines:**
\[ m_1 \cdot m_2 = \left(-\frac{1}{2}\right) \cdot \left(-\frac{2}{3}\right) = \frac{1}{3} \quad \text{(not perpendicular)} \]

2. **Between the first and third lines:**
\[ m_1 \cdot m_3 = \left(-\frac{1}{2}\right) \cdot (-k) = \frac{k}{2} = -1 \implies k = -2 \]

3. **Between the second and third lines:**
\[ m_2 \cdot m_3 = \left(-\frac{2}{3}\right) \cdot (-k) = \frac{2k}{3} = -1 \implies k = -\frac{3}{2} \]

### Step 3: Values of \( k \)

The values of \( k \) that make the triangle right-angled are:
- \( k_1 = -2 \)
- \( k_2 = -\frac{3}{2} \)

### Step 4: Calculate \( |(k_1 + k_2) / (k_1 - k_2)| \)

1. **Calculate \( k_1 + k_2 \):**
\[ k_1 + k_2 = -2 - \frac{3}{2} = -\frac{4}{2} - \frac{3}{2} = -\frac{7}{2} \]

2. **Calculate \( k_1 - k_2 \):**
\[ k_1 - k_2 = -2 + \frac{3}{2} = -\frac{4}{2} + \frac{3}{2} = -\frac{1}{2} \]

3. **Calculate \( \frac{k_1 + k_2}{k_1 - k_2} \):**
\[ \frac{k_1 + k_2}{k_1 - k_2} = \frac{-\frac{7}{2}}{-\frac{1}{2}} = \frac{7}{1} = 7 \]

4. **Final result:**
\[ |(k_1 + k_2) / (k_1 - k_2)| = |7| = 7 \]

### Final Answer:
The value of \( |(k_1 + k_2) / (k_1 - k_2)| \) is \( 7 \).