The equation to the pair of lines through the origin which are perpendicular to the lines represented by `ax^(2)+2hxy+by^(2)=0` is

Options

A. `bx^(2)+2hxy+ay^(2)=0`
B. `bx^(2)+2hxy-bh^(2)=0`
C. `bx^(2)-2hxy+ay^(2)=0`
D. none of these

Answer

To find the equation of the pair of lines through the origin that are perpendicular to the lines represented by the equation \( ax^2 + 2hxy + by^2 = 0 \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Given Equation**:
The equation \( ax^2 + 2hxy + by^2 = 0 \) represents a pair of straight lines through the origin. The coefficients \( a \), \( b \), and \( h \) are related to the slopes of these lines.

**Hint**: Recognize that the given equation represents two lines that intersect at the origin.

2. **Identify the Perpendicular Condition**:
We need to find the lines that are perpendicular to the lines represented by the given equation. For two lines to be perpendicular, the product of their slopes must equal -1.

**Hint**: Recall the condition for perpendicular lines in terms of their slopes.

3. **Formulate the Required Equation**:
The equation of the pair of lines through the origin can be expressed in the form \( px^2 + qy^2 - 2hxy = 0 \), where \( p \) and \( q \) are coefficients that we need to determine.

**Hint**: The general form for lines through the origin is \( Ax^2 + By^2 + Cxy = 0 \).

4. **Interchange Coefficients**:
To find the lines that are perpendicular, we interchange the coefficients of \( x^2 \) and \( y^2 \) from the original equation. Thus, we replace \( a \) with \( b \) and \( b \) with \( a \).

**Hint**: This step is crucial for ensuring that the new lines are perpendicular to the original lines.

5. **Change the Sign of \( h \)**:
We also need to change the sign of \( h \) to ensure that the new lines are perpendicular. Therefore, we replace \( h \) with \( -h \).

**Hint**: This adjustment is necessary to maintain the perpendicularity of the lines.

6. **Final Equation**:
After making these changes, the equation of the pair of lines through the origin that are perpendicular to the original lines is given by:
\[ bx^2 + ay^2 - 2(-h)xy = 0 \]
Simplifying this gives:
\[ bx^2 + ay^2 + 2hxy = 0 \]

**Hint**: Ensure that you write the final equation correctly, keeping track of the signs.

### Final Answer:
The equation of the pair of lines through the origin which are perpendicular to the lines represented by \( ax^2 + 2hxy + by^2 = 0 \) is:
\[ bx^2 + ay^2 + 2hxy = 0 \]