The lines bisecting the angle between the bisectors of the angles between the lines `ax^(2)+2hxy+by^(2)=0` are given by

Options

A. `(x^(2)-y^(2))/(a-b) = (xy)/(h)`
B. `(x^(2)+y^(2))/(a+b) = (xy)/(h)`
C. `(x^(2) - y^(2))/(a+b) = (xy)/(h)`
D. `(x^(2) - y^(2))/(a - b) = - (xy)/(h)`

Answer

To find the lines bisecting the angle between the bisectors of the angles formed by the lines given by the equation \( ax^2 + 2hxy + by^2 = 0 \), we can follow these steps:

### Step 1: Understand the given equation
The equation \( ax^2 + 2hxy + by^2 = 0 \) represents a pair of straight lines. The coefficients \( a \), \( b \), and \( h \) are related to the angles between these lines.

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

### Step 3: Rearrange the formula
We can rearrange the angle bisector equation:
\[ x^2 - y^2 = \frac{(a - b)xy}{h} \]

### Step 4: Identify values of \( a \), \( b \), and \( h \)
From the original equation \( ax^2 + 2hxy + by^2 = 0 \), we can identify:
- \( a = a \)
- \( b = b \)
- \( h = h \)

### Step 5: Substitute values into the angle bisector formula
Substituting the values of \( a \), \( b \), and \( h \) into the rearranged angle bisector formula gives:
\[ x^2 - y^2 = \frac{(a - b)xy}{h} \]

### Step 6: Simplify the equation
We can multiply both sides by \( h \) to eliminate the fraction:
\[ h(x^2 - y^2) = (a - b)xy \]

### Step 7: Rearranging the equation
Rearranging gives us:
\[ hx^2 - hy^2 - (a - b)xy = 0 \]

### Step 8: Identify the final equation
This is the equation of the lines bisecting the angle between the bisectors of the angles formed by the original pair of lines.

### Final Answer
The lines bisecting the angle between the bisectors of the angles formed by the lines \( ax^2 + 2hxy + by^2 = 0 \) are given by:
\[ hx^2 - hy^2 - (a - b)xy = 0 \]