Find the angle between the st. Lines :
`(a+b)x+(a-b)y=2ab` and `(a-b)x+(a+b)y=2ab` is ......................

Answer

To find the angle between the two straight lines given by the equations \((a+b)x + (a-b)y = 2ab\) and \((a-b)x + (a+b)y = 2ab\), we will follow these steps:

### Step 1: Identify the equations of the lines
The first line can be written as:
\[ (a+b)x + (a-b)y = 2ab \]
The second line can be written as:
\[ (a-b)x + (a+b)y = 2ab \]

### Step 2: Find the slopes of the lines
To find the slope of a line given in the form \(Ax + By = C\), we use the formula:
\[ \text{slope} = -\frac{A}{B} \]

#### For the first line:
- Coefficient of \(x\) (A) = \(a + b\)
- Coefficient of \(y\) (B) = \(a - b\)

Thus, the slope \(m_1\) of the first line is:
\[ m_1 = -\frac{a+b}{a-b} \]

#### For the second line:
- Coefficient of \(x\) (A) = \(a - b\)
- Coefficient of \(y\) (B) = \(a + b\)

Thus, the slope \(m_2\) of the second line is:
\[ m_2 = -\frac{a-b}{a+b} \]

### Step 3: Use the formula for the angle between two lines
The angle \(\theta\) between two lines with slopes \(m_1\) and \(m_2\) can be found using the formula:
\[ \tan \theta = \frac{m_1 - m_2}{1 + m_1 m_2} \]

### Step 4: Substitute the slopes into the formula
Substituting the values of \(m_1\) and \(m_2\):
\[ \tan \theta = \frac{-\frac{a+b}{a-b} - \left(-\frac{a-b}{a+b}\right)}{1 + \left(-\frac{a+b}{a-b}\right)\left(-\frac{a-b}{a+b}\right)} \]

### Step 5: Simplify the numerator
The numerator simplifies as follows:
\[ -\frac{a+b}{a-b} + \frac{a-b}{a+b} = -\frac{(a+b)(a+b) + (a-b)(a-b)}{(a-b)(a+b)} \]
This can be simplified to:
\[ -\frac{(a+b)^2 - (a-b)^2}{(a-b)(a+b)} \]

### Step 6: Simplify the denominator
The denominator simplifies as follows:
\[ 1 + \frac{(a+b)(a-b)}{(a-b)(a+b)} = 1 + 1 = 2 \]

### Step 7: Combine the results
Now we have:
\[ \tan \theta = \frac{-\frac{(a+b)^2 - (a-b)^2}{(a-b)(a+b)}}{2} \]

### Step 8: Final simplification
After simplifying, we find:
\[ \tan \theta = -\frac{(a+b)^2 - (a-b)^2}{2(a-b)(a+b)} \]
Calculating the squares and simplifying further gives:
\[ \tan \theta = -\frac{-4ab}{2} = -2ab \]

### Step 9: Find \(\theta\)
Thus, we have:
\[ \theta = \tan^{-1}(-2ab) \]

### Conclusion
The angle between the two lines is:
\[ \theta = \tan^{-1}(-2ab) \]