The angle between the lines represented by the equation `ax^(2)+xy+by^(2)=0` will be `45^(@)` if

Options

A. `a=1,b=6`
B. `a=1,b=-6`
C. `a=6,b=1`
D. `a=1,b=1`

Explanation

To solve the problem of finding the condition under which the angle between the lines represented by the equation \( ax^2 + xy + by^2 = 0 \) is \( 45^\circ \), we can follow these steps:

### Step 1: Identify the coefficients
The given equation is \( ax^2 + xy + by^2 = 0 \). Here, we can identify the coefficients:
- \( A = a \)
- \( B = 1 \) (since the coefficient of \( xy \) is 1)
- \( C = b \)

### Step 2: Use the angle formula
The formula for the tangent of the angle \( \theta \) between the two lines represented by the general quadratic equation \( Ax^2 + 2Hxy + By^2 = 0 \) is given by:

\[ \tan \theta = \frac{2\sqrt{H^2 - AB}}{A + B} \]

In our case, we have \( H = \frac{1}{2} \), \( A = a \), and \( B = b \). Therefore, we can substitute these values into the formula:

\[ \tan \theta = \frac{2\sqrt{\left(\frac{1}{2}\right)^2 - ab}}{a + b} \]

### Step 3: Set the angle to \( 45^\circ \)
We know that \( \tan 45^\circ = 1 \). Thus, we set the equation:

\[ 1 = \frac{2\sqrt{\left(\frac{1}{2}\right)^2 - ab}}{a + b} \]

### Step 4: Cross-multiply and simplify
Cross-multiplying gives us:

\[ a + b = 2\sqrt{\left(\frac{1}{2}\right)^2 - ab} \]

Squaring both sides results in:

\[ (a + b)^2 = 4\left(\frac{1}{4} - ab\right) \]

Expanding both sides:

\[ a^2 + 2ab + b^2 = 1 - 4ab \]

### Step 5: Rearranging the equation
Rearranging the equation gives us:

\[ a^2 + b^2 + 6ab = 1 \]

### Step 6: Conclusion
The condition for the angle between the lines represented by the equation \( ax^2 + xy + by^2 = 0 \) to be \( 45^\circ \) is:

\[ a^2 + b^2 + 6ab = 1 \]