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, we need to find the values of \( k \) such that the lines represented by the equations \( x + 2y = 3 \), \( 2x + 3y = 5 \), and \( k^2 x + ky = -1 \) form a right-angled triangle. We will then compute the expression \( \left| \frac{k_1 + k_2}{k_1 - k_2} \right| \).

### Step 1: Write the equations in slope-intercept form
1. **Equation 1**: \( x + 2y = 3 \)
- Rearranging gives \( 2y = -x + 3 \) or \( y = -\frac{1}{2}x + \frac{3}{2} \)
- Slope \( m_1 = -\frac{1}{2} \)

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

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

### Step 2: Determine conditions for right angles
For the triangle to be right-angled, the product of the slopes of two lines must equal -1. We will check the combinations of the slopes:

1. **Condition 1**: \( m_1 \cdot m_3 = -1 \)
\[ -\frac{1}{2} \cdot (-k) = -1 \implies \frac{k}{2} = -1 \implies k = -2 \]

2. **Condition 2**: \( m_2 \cdot m_3 = -1 \)
\[ -\frac{2}{3} \cdot (-k) = -1 \implies \frac{2k}{3} = -1 \implies k = -\frac{3}{2} \]

### Step 3: Values of \( k \)
We have found two values:
- \( k_1 = -2 \)
- \( k_2 = -\frac{3}{2} \)

### Step 4: Calculate \( \left| \frac{k_1 + k_2}{k_1 - k_2} \right| \)
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**:
\[ \left| \frac{k_1 + k_2}{k_1 - k_2} \right| = |7| = 7 \]

### Final Answer
The value of \( \left| \frac{k_1 + k_2}{k_1 - k_2} \right| \) is \( \boxed{7} \).