The equation `6x^(2)-xy-12y^(2)-8x+29y-14=0` represents a pair of lines and angle between them is

Options

A. `tan^(-1)(-17/6)`
B. `tan^(-1)(3/4)`
C. `pi//4`
D. `pi//3`

Answer

To find the angle between the pair of lines represented by the equation \(6x^2 - xy - 12y^2 - 8x + 29y - 14 = 0\), we can follow these steps:

### Step 1: Identify coefficients
The general form of the equation of a pair of straight lines is given by:
\[ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \]
From the equation \(6x^2 - xy - 12y^2 - 8x + 29y - 14 = 0\), we can identify:
- \(A = 6\)
- \(B = -1\)
- \(C = -12\)
- \(D = -8\)
- \(E = 29\)
- \(F = -14\)

### Step 2: Check the condition for a pair of lines
For the equation to represent a pair of lines, the condition is:
\[ B^2 - 4AC = 0 \]
Calculating:
\[ (-1)^2 - 4 \cdot 6 \cdot (-12) = 1 + 288 = 289 \quad (\text{which is } > 0) \]
Since \(B^2 - 4AC > 0\), the equation represents a pair of lines.

### Step 3: Calculate \(h\), \(a\), and \(b\)
We need to calculate:
- \(h = \frac{B}{2} = \frac{-1}{2} = -\frac{1}{2}\)
- \(a = A = 6\)
- \(b = C = -12\)

### Step 4: Use the angle formula
The angle \(\theta\) between the two lines can be calculated using the formula:
\[ \tan \theta = \frac{2\sqrt{h^2 - ab}}{a + b} \]
Substituting the values:
- \(h^2 = \left(-\frac{1}{2}\right)^2 = \frac{1}{4}\)
- \(ab = 6 \cdot (-12) = -72\)

Now substituting into the formula:
\[ \tan \theta = \frac{2\sqrt{\frac{1}{4} - (-72)}}{6 + (-12)} \]
Calculating:
\[ \tan \theta = \frac{2\sqrt{\frac{1}{4} + 72}}{-6} \]
\[ = \frac{2\sqrt{\frac{1}{4} + \frac{288}{4}}}{-6} = \frac{2\sqrt{\frac{289}{4}}}{-6} = \frac{2 \cdot \frac{17}{2}}{-6} = \frac{17}{-6} \]
Thus, we have:
\[ \tan \theta = -\frac{17}{6} \]

### Step 5: Find \(\theta\)
Now, we can find the angle:
\[ \theta = \tan^{-1}\left(-\frac{17}{6}\right) \]

### Conclusion
The angle between the pair of lines represented by the equation is \(\tan^{-1}\left(-\frac{17}{6}\right)\).

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