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

Options

A. `px^(2)+2xy-py^(2)=0`
B. `px^(2)+2xy+py^(2)=0`
C. `x^(2)-2pxy+y^(2)=0`
D. None of these

Answer

To solve the problem of finding 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, we can follow these steps:

### Step-by-Step Solution:

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

2. **Use the formula for the angle bisectors**:
The formula for the angle bisectors of the lines represented by the equation \( ax^2 + 2hxy + by^2 = 0 \) is given by:
\[ \frac{x^2 - y^2}{a - b} = \frac{xy}{h} \]

3. **Substitute the values of \( a \), \( b \), and \( h \)**:
Substituting \( a = 1 \), \( b = -1 \), and \( h = -p \) into the formula:
\[ \frac{x^2 - y^2}{1 - (-1)} = \frac{xy}{-p} \]
Simplifying this gives:
\[ \frac{x^2 - y^2}{2} = \frac{xy}{-p} \]

4. **Cross-multiply to eliminate the fractions**:
Cross-multiplying gives:
\[ -p(x^2 - y^2) = 2xy \]

5. **Rearranging the equation**:
Rearranging the equation leads to:
\[ -px^2 + py^2 + 2xy = 0 \]
or, multiplying through by -1:
\[ px^2 - 2xy - py^2 = 0 \]

6. **Final equation**:
Thus, the equation of the bisectors of the angle between the lines in the new position after rotation is:
\[ px^2 - 2xy - py^2 = 0 \]