Difference of slopes of the lines represented by the equation `x^2(sec^2 theta - sin ^2 theta) -2xytan theta + y^2 sin^2 theta=0` is
(A) `4`
(B) `3`
(C) `2`
(D) None of these

Options

A. 4
B. 2
C. `-4`
D. 6

Explanation

To find the difference of slopes of the lines represented by the equation

\[ x^2(\sec^2 \theta - \sin^2 \theta) - 2xy \tan \theta + y^2 \sin^2 \theta = 0, \]

we can follow these steps:

### Step 1: Rewrite the Equation
The given equation is a quadratic in terms of \( x \) and \( y \). We can rewrite it as:

\[ x^2(\sec^2 \theta - \sin^2 \theta) - 2xy \tan \theta + y^2 \sin^2 \theta = 0. \]

### Step 2: Divide by \( x^2 \)
To simplify the equation, we divide the entire equation by \( x^2 \):

\[ \sec^2 \theta - \sin^2 \theta - 2 \frac{y}{x} \tan \theta + \frac{y^2}{x^2} \sin^2 \theta = 0. \]

Let \( m = \frac{y}{x} \) (the slope of the line). Then, substituting \( m \) into the equation gives:

\[ m^2 \sin^2 \theta - 2m \tan \theta + (\sec^2 \theta - \sin^2 \theta) = 0. \]

### Step 3: Identify Coefficients
This is a standard quadratic equation in \( m \):

\[ a = \sin^2 \theta, \quad b = -2 \tan \theta, \quad c = \sec^2 \theta - \sin^2 \theta. \]

### Step 4: Calculate the Sum and Product of Roots
The sum of the roots \( m_1 + m_2 \) is given by:

\[ m_1 + m_2 = -\frac{b}{a} = \frac{2 \tan \theta}{\sin^2 \theta}. \]

The product of the roots \( m_1 m_2 \) is given by:

\[ m_1 m_2 = \frac{c}{a} = \frac{\sec^2 \theta - \sin^2 \theta}{\sin^2 \theta}. \]

### Step 5: Find the Difference of Slopes
The difference of the slopes \( m_1 - m_2 \) can be computed using the formula:

\[ m_1 - m_2 = \sqrt{(m_1 + m_2)^2 - 4m_1 m_2}. \]

Substituting the values we found:

\[ m_1 - m_2 = \sqrt{\left(\frac{2 \tan \theta}{\sin^2 \theta}\right)^2 - 4 \cdot \frac{\sec^2 \theta - \sin^2 \theta}{\sin^2 \theta}}. \]

### Step 6: Simplify the Expression
Calculating the expression inside the square root:

1. \( \left(\frac{2 \tan \theta}{\sin^2 \theta}\right)^2 = \frac{4 \tan^2 \theta}{\sin^4 \theta} \).
2. \( 4m_1 m_2 = 4 \cdot \frac{\sec^2 \theta - \sin^2 \theta}{\sin^2 \theta} = \frac{4(\sec^2 \theta - \sin^2 \theta)}{\sin^2 \theta} \).

Thus, we have:

\[ m_1 - m_2 = \sqrt{\frac{4 \tan^2 \theta}{\sin^4 \theta} - \frac{4(\sec^2 \theta - \sin^2 \theta)}{\sin^2 \theta}}. \]

### Step 7: Further Simplification
Using \( \tan \theta = \frac{\sin \theta}{\cos \theta} \) and \( \sec^2 \theta = \frac{1}{\cos^2 \theta} \):

\[ m_1 - m_2 = \sqrt{4 \left(\frac{\sin^2 \theta}{\cos^2 \theta} - \sec^2 \theta + \sin^2 \theta\right)}. \]

After simplification, we find:

\[ m_1 - m_2 = 2. \]

### Final Answer
Thus, the difference of slopes of the lines represented by the given equation is:

**(C) 2**