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

Answer

To find the angle between the two given straight lines, we will follow these steps:

### Step 1: Write the equations of the lines in slope-intercept form
The equations of the lines are:
1. \((a+b)x + (a-b)y = 2ab\)
2. \((a-b)x + (a+b)y = 2ab\)

We need to rearrange these equations to find their slopes.

### Step 2: Rearranging the first equation
Starting with the first equation:
\[ (a+b)x + (a-b)y = 2ab \]
Rearranging for \(y\):
\[ (a-b)y = 2ab - (a+b)x \]
\[ y = \frac{2ab - (a+b)x}{(a-b)} \]
This can be rewritten as:
\[ y = -\frac{(a+b)}{(a-b)}x + \frac{2ab}{(a-b)} \]
From this, we can identify the slope \(m_1\):
\[ m_1 = -\frac{(a+b)}{(a-b)} \]

### Step 3: Rearranging the second equation
Now, let's rearrange the second equation:
\[ (a-b)x + (a+b)y = 2ab \]
Rearranging for \(y\):
\[ (a+b)y = 2ab - (a-b)x \]
\[ y = \frac{2ab - (a-b)x}{(a+b)} \]
This can also be rewritten as:
\[ y = -\frac{(a-b)}{(a+b)}x + \frac{2ab}{(a+b)} \]
From this, we can identify the slope \(m_2\):
\[ m_2 = -\frac{(a-b)}{(a+b)} \]

### Step 4: Finding the angle between the two lines
The formula to find the angle \(\theta\) between two lines with slopes \(m_1\) and \(m_2\) is given by:
\[ \tan \theta = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right| \]

Substituting the values of \(m_1\) and \(m_2\):
\[ \tan \theta = \left| \frac{-\frac{(a+b)}{(a-b)} + \frac{(a-b)}{(a+b)}}{1 + \left(-\frac{(a+b)}{(a-b)}\right)\left(-\frac{(a-b)}{(a+b)}\right)} \right| \]

### Step 5: Simplifying the expression
Calculating the numerator:
\[ -\frac{(a+b)}{(a-b)} + \frac{(a-b)}{(a+b)} = -\frac{(a+b)(a+b) + (a-b)(a-b)}{(a-b)(a+b)} \]
This simplifies to:
\[ -\frac{(a+b)^2 - (a-b)^2}{(a-b)(a+b)} = -\frac{(a^2 + 2ab + b^2) - (a^2 - 2ab + b^2)}{(a-b)(a+b)} = -\frac{4ab}{(a-b)(a+b)} \]

Calculating the denominator:
\[ 1 + \frac{(a+b)(a-b)}{(a-b)(a+b)} = 1 + 1 = 2 \]

Thus, we have:
\[ \tan \theta = \left| \frac{-\frac{4ab}{(a-b)(a+b)}}{2} \right| = \frac{2ab}{(a-b)(a+b)} \]

### Step 6: Final result
Therefore, the angle \(\theta\) between the two lines is given by:
\[ \theta = \tan^{-1}\left(\frac{2ab}{(a-b)(a+b)}\right) \]