The combined equation of straight lines that can be obtained by reflecting the lines y=x-2 in the y-axis is
y2+x2+4x+4=0
y2+x2−4x+4=0
y2−x2+4x−4=0
y2−x2−4x−4=0
If we reflect y=x-2 in y-axis, it will becomes y=|−x−2|=|x+2|. The reflected lines are y=x+2, y=−x−2. Their combined equation is (y−x−2)(y+x+2)=0⇒y2−(x+2)2=0
⇒y2−x2−4x−4=0