The acute angle between the lines `ax+by+c=0` and `(a+b)x=(a-b)y,a!=b` is

Options

A. `15^(@)`
B. `30^(@)`
C. `45^(@)`
D. `60^(@)`

Answer

To find the acute angle between the lines given by the equations \( ax + by + c = 0 \) and \( (a + b)x = (a - b)y \) (where \( a \neq b \)), we will follow these steps:

### Step 1: Find the slopes of the lines

1. **First Line**: The equation is \( ax + by + c = 0 \).
- Rearranging gives us \( by = -ax - c \) or \( y = -\frac{a}{b}x - \frac{c}{b} \).
- The slope \( m_1 \) of the first line is \( -\frac{a}{b} \).

2. **Second Line**: The equation is \( (a + b)x = (a - b)y \).
- Rearranging gives us \( y = \frac{(a + b)}{(a - b)}x \).
- The slope \( m_2 \) of the second line is \( \frac{(a + b)}{(a - b)} \).

### Step 2: Use the formula for the angle between two lines

The formula for the tangent of the angle \( \theta \) between two lines with slopes \( m_1 \) and \( m_2 \) is given by:

\[ \tan \theta = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right| \]

### Step 3: Substitute the slopes into the formula

Substituting \( m_1 = -\frac{a}{b} \) and \( m_2 = \frac{(a + b)}{(a - b)} \):

\[ \tan \theta = \left| \frac{-\frac{a}{b} - \frac{(a + b)}{(a - b)}}{1 + \left(-\frac{a}{b}\right) \left(\frac{(a + b)}{(a - b)}\right)} \right| \]

### Step 4: Simplify the expression

1. **Numerator**:
- Combine the fractions:
\[ -\frac{a}{b} - \frac{(a + b)}{(a - b)} = -\frac{a(a - b) + b(a + b)}{b(a - b)} = -\frac{a^2 - ab + ab + b^2}{b(a - b)} = -\frac{a^2 + b^2}{b(a - b)} \]

2. **Denominator**:
- Calculate:
\[ 1 - \frac{a(a + b)}{b(a - b)} = \frac{b(a - b) - a(a + b)}{b(a - b)} = \frac{ba - b^2 - a^2 - ab}{b(a - b)} = \frac{-a^2 + b^2}{b(a - b)} \]

3. **Putting it all together**:
\[ \tan \theta = \left| \frac{-\frac{a^2 + b^2}{b(a - b)}}{\frac{-a^2 + b^2}{b(a - b)}} \right| = \left| \frac{a^2 + b^2}{-a^2 + b^2} \right| \]

### Step 5: Find the angle

Since \( \tan \theta = \frac{a^2 + b^2}{b^2 - a^2} \), we can analyze the situation further.

For the acute angle, we can conclude that \( \theta = 45^\circ \) when \( a^2 + b^2 \) and \( b^2 - a^2 \) are equal, which simplifies to \( \tan \theta = 1 \).

### Final Answer

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

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