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

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:
- \(a = 1\)
- \(h = \sec \theta\)
- \(b = 1\)

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

### Step 3: Substitute into the formula
Now substituting these values into the formula:
\[ \tan \alpha = \frac{2\sqrt{\sec^2 \theta - 1}}{1 + 1} \]
This simplifies to:
\[ \tan \alpha = \frac{2\sqrt{\sec^2 \theta - 1}}{2} \]
\[ \tan \alpha = \sqrt{\sec^2 \theta - 1} \]

### Step 4: Simplify using trigonometric identities
Using the identity \(\sec^2 \theta - 1 = \tan^2 \theta\), we can rewrite:
\[ \tan \alpha = \sqrt{\tan^2 \theta} \]
This gives us:
\[ \tan \alpha = |\tan \theta| \]

### Step 5: Determine the angle
Since \(\tan \alpha = \tan \theta\), we conclude that:
\[ \alpha = \theta \]

### Final Answer
Thus, the angle between the lines is:
\[ \alpha = \theta \]

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