Angle between the pair of straight lines `x^(2) - xy - 6y^(2) - 2x + 11y - 3 = 0` is

Options

A. `45^(@), 135^(@)`
B. `tan^(-1) 2, pi - tan^(-1) 2`
C. `tan^(-1) 3, pi - tan^(-1) 3`
D. None of these

Answer

To find the angle between the pair of straight lines given by the equation \(x^2 - xy - 6y^2 - 2x + 11y - 3 = 0\), we can follow these steps:

### Step 1: Identify the coefficients
The general form of the equation of a pair of straight lines is given by:
\[ ax^2 + by^2 + 2hxy + 2gx + 2fy + c = 0 \]
From the given equation, we can identify the coefficients:
- \(a = 1\) (coefficient of \(x^2\))
- \(b = -6\) (coefficient of \(y^2\))
- \(h = -\frac{1}{2}\) (coefficient of \(xy\) is \(-1\), so \(2h = -1\) implies \(h = -\frac{1}{2}\))
- \(g = -1\) (coefficient of \(x\) is \(-2\), so \(2g = -2\) implies \(g = -1\))
- \(f = \frac{11}{2}\) (coefficient of \(y\) is \(11\), so \(2f = 11\) implies \(f = \frac{11}{2}\))
- \(c = -3\)

### Step 2: Use the formula for the angle between the lines
The formula for the tangent of the angle \(\theta\) between the pair of straight lines is given by:
\[ \tan \theta = \left| \frac{2\sqrt{h^2 - ab}}{a + b} \right| \]

### Step 3: Calculate \(h^2 - ab\)
Now we need to calculate \(h^2 - ab\):
\[ h^2 = \left(-\frac{1}{2}\right)^2 = \frac{1}{4} \]
\[ ab = 1 \cdot (-6) = -6 \]
\[ h^2 - ab = \frac{1}{4} - (-6) = \frac{1}{4} + 6 = \frac{1}{4} + \frac{24}{4} = \frac{25}{4} \]

### Step 4: Calculate \(a + b\)
Now calculate \(a + b\):
\[ a + b = 1 - 6 = -5 \]

### Step 5: Substitute into the formula
Now substitute \(h^2 - ab\) and \(a + b\) into the formula for \(\tan \theta\):
\[ \tan \theta = \left| \frac{2\sqrt{\frac{25}{4}}}{-5} \right| = \left| \frac{2 \cdot \frac{5}{2}}{-5} \right| = \left| \frac{5}{-5} \right| = 1 \]

### Step 6: Determine the angle \(\theta\)
Since \(\tan \theta = 1\), we have:
\[ \theta = 45^\circ \text{ or } 225^\circ \]
The angle between the lines is typically taken as the acute angle, so:
\[ \theta = 45^\circ \]

### Final Answer
The angle between the pair of straight lines is \(45^\circ\).