Pair Of Straight Lines Perpendicularity

Question

Find the condition that one of the lines given by $ax^2+2hxy+by^2=0$ may be perpendicular to one of the lines given by $Ax^2+2Hxy+By^2=0$. I tried out like $m_1 + m_2 = -2h/b$ and $m_1\cdot m_2=a/b$ for both equations. Now how to proceed?

Answer 1

Let $y=mx$ be equation of one of the straight lines $ax^2+2hxy+by^2=0$

Setting $y=mx,bm^2+2hm+a=0\ \ \ \ (1)$

Similarly let $y=-\dfrac xm\iff x=-my$ be equation of one of the straight lines $Ax^2+2Hxy+By^2=0$

Setting $x=-my,A(-my)^2+2H(-my)y+By^2=0\implies Am^2-2Hm+B=0\ \ \ \ (2)$

We need to have at least one root common for $(1),(2)$

Solve for $m,m^2$ and use the identity $m^2=(m)^2$ to eliminate $m$