Question

Equation of the image of the pair of lines \[ ax^2+2hxy+by^2=0 \] with respect to \(y=0\) is

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

Hint:

For reflection about the \(x\)-axis, \[ \boxed{(x,y)\rightarrow(x,-y).} \] Only the terms containing an odd power of \(y\) change their sign.

Answer 1

The Correct Option is B

Step 1: Use reflection about the \(x\)-axis. Reflection with respect to \[ y=0 \] changes \[ (x,y) \longrightarrow (x,-y). \] Hence, replace \[ y \] by \[ -y \] in the given equation.

Step 2: Substitute \(y=-y\). The given equation is \[ ax^2+2hxy+by^2=0. \] Replacing \[ y \] by \[ -y, \] we get \[ ax^2+2hx(-y)+b(-y)^2=0. \] Therefore, \[ ax^2-2hxy+by^2=0. \]

Step 3: Write the final equation. Hence, the required image is \[ \boxed{ax^2-2hxy+by^2=0.} \] Thus, \[ \boxed{(B)} \] is the correct answer.