Find the angle between the lines represented by `x^2+2x ysectheta+y^2=0`

Options

A. `4 theta`
B. `2 theta`
C. `theta`
D. none of these

Explanation

To find the angle between the lines represented by the equation \( x^2 + 2xy \sec \theta + y^2 = 0 \), 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:
\[ ax^2 + 2hxy + by^2 = 0 \]
From the given equation \( x^2 + 2xy \sec \theta + y^2 = 0 \), we can identify:
- \( a = 1 \)
- \( b = 1 \)
- \( h = \sec \theta \)

### Step 2: Use the formula for the tangent of the angle between the lines
The formula for the tangent of the angle \( \alpha \) between the two lines is:
\[ \tan \alpha = \frac{2\sqrt{h^2 - ab}}{a + b} \]
Substituting the values of \( a \), \( b \), and \( h \):
- \( h^2 = (\sec \theta)^2 = \sec^2 \theta \)
- \( ab = 1 \times 1 = 1 \)

Now substituting these into the formula:
\[ \tan \alpha = \frac{2\sqrt{\sec^2 \theta - 1}}{1 + 1} \]

### Step 3: Simplify the expression
We know that \( \sec^2 \theta - 1 = \tan^2 \theta \). Thus:
\[ \tan \alpha = \frac{2\sqrt{\tan^2 \theta}}{2} = \tan \theta \]

### Step 4: Conclude the angle between the lines
Since \( \tan \alpha = \tan \theta \), it follows that:
\[ \alpha = \theta \]
Thus, the angle between the lines is \( \theta \).

### Final Answer
The angle between the lines represented by the equation \( x^2 + 2xy \sec \theta + y^2 = 0 \) is \( \theta \).

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