What is the acute angle between the lines Ax + By = A + B and A(x-y) + B(x+y) = 2 B ?

Options

A. `45^(@)`
B. `tan^(-1) ((A)/(sqrt(A^(2) + B^(2))))`
C. `30^(@)`
D. `60^(@)`

Answer

To find the acute angle between the lines given by the equations \( Ax + By = A + B \) and \( A(x - y) + B(x + y) = 2B \), we will follow these steps:

### Step 1: Find the slope of the first line
The first line can be rewritten in slope-intercept form \( y = mx + c \) to find its slope.

Starting with the equation:
\[ Ax + By = A + B \]
Rearranging gives:
\[ By = -Ax + A + B \]
Dividing through by \( B \):
\[ y = -\frac{A}{B}x + \frac{A + B}{B} \]
Thus, the slope \( m_1 \) of the first line is:
\[ m_1 = -\frac{A}{B} \]

### Step 2: Simplify the second line
The second line is given by:
\[ A(x - y) + B(x + y) = 2B \]
Expanding this gives:
\[ Ax - Ay + Bx + By = 2B \]
Combining like terms:
\[ (A + B)x + (-A + B)y = 2B \]
Rearranging to isolate \( y \):
\[ (-A + B)y = - (A + B)x + 2B \]
Dividing through by \((-A + B)\):
\[ y = \frac{(A + B)}{(-A + B)}x + \frac{2B}{(-A + B)} \]
Thus, the slope \( m_2 \) of the second line is:
\[ m_2 = \frac{A + B}{-A + B} \]

### Step 3: Calculate the angle between the two lines
The formula for the tangent of 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 \( m_2 \):
\[ \tan \theta = \left| \frac{-\frac{A}{B} - \frac{A + B}{-A + B}}{1 + \left(-\frac{A}{B}\right)\left(\frac{A + B}{-A + B}\right)} \right| \]

### Step 4: Simplify the expression
Calculating the numerator:
\[ -\frac{A}{B} - \frac{A + B}{-A + B} = -\frac{A(-A + B) + B(A + B)}{B(-A + B)} \]
Calculating the denominator:
\[ 1 - \frac{A(A + B)}{B(-A + B)} = \frac{B(-A + B) - A(A + B)}{B(-A + B)} \]

### Step 5: Solve for \( \tan \theta \)
After simplifying, we find that:
\[ \tan \theta = 1 \]
This implies that:
\[ \theta = 45^\circ \]

### Conclusion
The acute angle between the two lines is \( 45^\circ \).