Question

Find the equation to the pair of lines through the origin which are perpendicular to the lines represented by $a{x}^2+2hxy+b{y}^2=0$.

Answer 1

If the lines represented by $a{x}^2+2hxy+b{y}^2=0$,
be $y-m_{1}x=0$ and $y-m_{2}x=0$ then
$m_1+m_2=-\frac{2h}{b}$ and $m_1{m}_2=\frac{a}{b}$
The lines perpendicular to them and passing through origin
will be
$y=-\frac{1}{m_1}x$ and $y=-\frac{1}{m_2}x$.
Their combined equation is
$(m_{1}y+x)(m_{2}y+x)=0$
or $m_1{m}_2{y}^2+xy(m_1+m_2)+x^2=0$
or $\frac{a}{b}\cdot y^2+xy(-\frac{2h}{b})+x^2=0$
or $b{x}^2-2hxy+a{y}^2=0$
Rule : Interchange the coefficient of $x^2$ and $y^2$ and change the sign of the term of $xy$.