Geo3.1

Find the equation to the pair of lines passing through the origin and perpendicular to the pair $ax^2+2hxy+by^2=0\,$ is $bx^2-2hxy+ay^2=0\,$

Let the lines be $lx_1+m_1y=0,l_2x+m_2y=0\,$ ,let these equations be 1 and 2.

Hence $(lx_1+m_1y)(l_2x+m_2y)\equiv ax^2+2hxy+by^2\,$

Therefore, $l_1 l_2=a,m_1 m_2=b,l_1 m_2+l_2 m_1=2h\,$

Equations to the lines passing thro' the origin and perpendicular to the 1 and 2 are

$m_1 x-l_1 y=0,m_2 x-l_2 y=0\,$

Therefore,the equation to pair of perpendicular lines is $(m_1 x-l_1 y)(m_2 x-l_2 y)=0\,$

$m_1 m_2x^2-xy(l_1 m_2+l_2 m_1)+l_1 l_2 y^2=0\,$

$bx^2-2hxy+ay^2=0\,$