Find the angle between the lines repersented by the equation `x^2-2pxy+y^2=0`

Answer

To find the angle between the lines represented by the equation \(x^2 - 2pxy + y^2 = 0\), we can follow these steps:

### Step 1: Write down the given equation
The given equation is:
\[ x^2 - 2pxy + y^2 = 0 \]

### Step 2: Identify coefficients
We can compare this equation with the standard form of the equation of two lines, which is:
\[ ax^2 + 2hxy + by^2 = 0 \]
From the equation \(x^2 - 2pxy + y^2 = 0\), we identify:
- \(a = 1\)
- \(h = -p\)
- \(b = 1\)

### Step 3: Use the formula for the angle between two lines
The formula for the tangent of the angle \(\theta\) between the two lines is given by:
\[ \tan \theta = \frac{2\sqrt{h^2 - ab}}{a + b} \]
Substituting the values of \(a\), \(b\), and \(h\):
\[ \tan \theta = \frac{2\sqrt{(-p)^2 - (1)(1)}}{1 + 1} \]

### Step 4: Simplify the expression
Calculating the values:
- \(h^2 = (-p)^2 = p^2\)
- \(ab = 1 \cdot 1 = 1\)

Thus, we have:
\[ \tan \theta = \frac{2\sqrt{p^2 - 1}}{2} \]
This simplifies to:
\[ \tan \theta = \sqrt{p^2 - 1} \]

### Step 5: Find the angle \(\theta\)
Now, we can express \(\theta\) in terms of the tangent:
\[ \theta = \tan^{-1}(\sqrt{p^2 - 1}) \]

### Final Answer
Therefore, the angle between the two lines is:
\[ \theta = \tan^{-1}(\sqrt{p^2 - 1}) \]

---