Question

Find the combined equation of the images of the pair of lines represented by $a{x}^2+2hxy+b{y}^2=0$ in the line mirror $y=0$.

Answer 1

Let the given lines be $y=m_{1}x,y=m_{2}x$ where
$m_1=\tan {\theta }_1,m_2=\tan {\theta }_2$
The image of $y=m_{1}x$ about $y=0$ i.e., axis of
$x$ will make an angle $-{\theta }_1$ with $x-$axis
and hence its equation is $y=\tan (-{\theta }_1)x$
or $y=-\tan {\theta }_{1}x$ or $y+m_{1}x=0$
$\therefore$ Combined equation is
$(y+m_{1}x)(y+m_{2}x)=0$
or $y^2+xy(m_1+m_2)+m_1{m}_2{x}^2=0$
or $b{y}^2-2hxy+a{y}^2=0$