If the lines represented by `x^(2)-2pxy-y^(2)=0` are rotated about the origin through an angle `theta`, one clockwise direction and other in anti-clockwise direction, then the equation of the bisectors of the angle between the lines in the new position is

Answer

To find the equation of the bisectors of the angle between the lines represented by the equation \(x^2 - 2pxy - y^2 = 0\) after rotating them about the origin through an angle \(\theta\), we will follow these steps:

### Step-by-Step Solution:

1. **Identify the coefficients**: The given equation is \(x^2 - 2pxy - y^2 = 0\). We can identify the coefficients \(a\), \(b\), and \(h\) from the general form \(ax^2 + 2hxy + by^2 = 0\):
- \(a = 1\)
- \(b = -1\)
- \(h = -p\)

2. **Use the formula for the bisectors**: The equation of the bisectors of the angle between the lines is given by:
\[ \frac{x^2 - y^2}{a - b} = \frac{xy}{h} \]
Substituting the values of \(a\), \(b\), and \(h\):
- \(a - b = 1 - (-1) = 2\)
- \(h = -p\)

Thus, the equation becomes:
\[ \frac{x^2 - y^2}{2} = \frac{xy}{-p} \]

3. **Cross-multiply to eliminate the fraction**: We can cross-multiply to simplify the equation:
\[ x^2 - y^2 = -\frac{2xy}{p} \]

4. **Rearranging the equation**: Rearranging gives:
\[ -px^2 + py^2 + 2xy = 0 \]

5. **Multiply through by -1**: To make the equation neater, we can multiply the entire equation by -1:
\[ px^2 - py^2 - 2xy = 0 \]

This is the equation of the bisectors of the angle between the lines in their new position after rotation.

### Final Answer:
The equation of the bisectors of the angle between the lines after rotation is:
\[ px^2 - py^2 - 2xy = 0 \]