Consider the equation of a pair of straight lines as `2x^(2)-10xy+12y^(2)+5x-16y-3=0`. The angles between the lines is `theta` . Then the value of `tan theta` is

Options

A. `1//5`
B. `2//9`
C. `1//7`
D. `3//4`

Explanation

To find the value of \( \tan \theta \) for the given equation of a pair of straight lines, we will follow these steps:

### Step 1: Identify the coefficients from the given equation
The given equation is:
\[ 2x^2 - 10xy + 12y^2 + 5x - 16y - 3 = 0 \]
We can compare this with the standard form of the equation of a pair of straight lines:
\[ ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0 \]
From the given equation, we can identify:
- \( a = 2 \)
- \( 2h = -10 \) (thus \( h = -5 \))
- \( b = 12 \)
- \( 2g = 5 \) (thus \( g = \frac{5}{2} \))
- \( 2f = -16 \) (thus \( f = -8 \))
- \( c = -3 \)

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

### Step 3: Substitute the values into the formula
Now we substitute the values we found:
- \( a = 2 \)
- \( b = 12 \)
- \( h = -5 \)

Calculating \( h^2 - ab \):
\[ h^2 = (-5)^2 = 25 \]
\[ ab = 2 \times 12 = 24 \]
Thus,
\[ h^2 - ab = 25 - 24 = 1 \]

Now substituting into the formula:
\[ \tan \theta = \frac{2\sqrt{1}}{2 + 12} = \frac{2 \cdot 1}{14} = \frac{2}{14} = \frac{1}{7} \]

### Step 4: Conclusion
The value of \( \tan \theta \) is:
\[ \tan \theta = \frac{1}{7} \]