The image of the pair of lines represented by `ax^(2)+2hxy+by^(2)=0`by the line mirror `y=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. `ax^(2)-2hxy+by^(2)=0`

Explanation

To find the image of the pair of lines represented by the equation \( ax^2 + 2hxy + by^2 = 0 \) with respect to the line \( y = 0 \) (the x-axis), we can follow these steps:

### Step 1: Understand the Reflection
The reflection of a point \((x, y)\) across the line \(y = 0\) is given by the point \((x, -y)\). This means that for any point on the lines represented by the equation, we will replace \(y\) with \(-y\).

### Step 2: Substitute \(y\) with \(-y\)
Starting with the equation of the pair of lines:
\[ ax^2 + 2hxy + by^2 = 0 \]
we substitute \(y\) with \(-y\):
\[ ax^2 + 2hx(-y) + b(-y)^2 = 0 \]

### Step 3: Simplify the Equation
Now simplify the equation:
\[ ax^2 - 2hxy + by^2 = 0 \]

### Step 4: Write the Final Result
The equation \( ax^2 - 2hxy + by^2 = 0 \) represents the image of the original pair of lines with respect to the line \(y = 0\).

### Final Answer
Thus, the image of the pair of lines represented by \( ax^2 + 2hxy + by^2 = 0 \) with respect to the line \( y = 0 \) is:
\[ ax^2 - 2hxy + by^2 = 0 \]