The combined equation of the pair of lines through the origin and perpendicular to the pair of lines given by `ax^(2)+2hxy+by^(2)=0`, is

Options

A. `ax^(2)-2hxy+by^(2)=0`
B. `bx^(2)+2hxy+ay^(2)=0`
C. `bx^(2)-2hxy+ay^(2)=0`
D. `bx^(2)+2hxy-ay^(2)=0`

Explanation

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

### Step 1: Identify the slopes of the given lines
The given equation \( ax^2 + 2hxy + by^2 = 0 \) represents a pair of lines through the origin. The slopes \( m_1 \) and \( m_2 \) of these lines can be found using the relationships:
- \( m_1 + m_2 = -\frac{2h}{b} \) (Equation 1)
- \( m_1 m_2 = \frac{a}{b} \) (Equation 2)

### Step 2: Write the equations of the lines through the origin
The equations of the lines through the origin with slopes \( m_1 \) and \( m_2 \) can be expressed as:
- \( y = m_1 x \)
- \( y = m_2 x \)

### Step 3: Find the slopes of the perpendicular lines
For lines that are perpendicular to the lines with slopes \( m_1 \) and \( m_2 \), the slopes of the new lines will be \( -\frac{1}{m_1} \) and \( -\frac{1}{m_2} \).

### Step 4: Write the equations of the perpendicular lines
The equations of the lines through the origin and perpendicular to the original lines are:
- \( y + \frac{1}{m_1} x = 0 \)
- \( y + \frac{1}{m_2} x = 0 \)

### Step 5: Combine the equations of the perpendicular lines
To find the combined equation of these lines, we multiply the two equations:
\[ (y + \frac{1}{m_1} x)(y + \frac{1}{m_2} x) = 0 \]
Expanding this gives:
\[ y^2 + \left(\frac{1}{m_1} + \frac{1}{m_2}\right)xy + \frac{1}{m_1 m_2}x^2 = 0 \]

### Step 6: Substitute values from Equations 1 and 2
From Equation 1, we have:
\[ \frac{1}{m_1} + \frac{1}{m_2} = \frac{m_1 + m_2}{m_1 m_2} = \frac{-\frac{2h}{b}}{\frac{a}{b}} = -\frac{2h}{a} \]
From Equation 2, we have:
\[ m_1 m_2 = \frac{a}{b} \]
Thus, substituting these into the combined equation gives:
\[ y^2 - \frac{2h}{a}xy + \frac{b}{a}x^2 = 0 \]
Multiplying through by \( a \) to eliminate the fraction results in:
\[ ay^2 - 2hxy + bx^2 = 0 \]

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