Q.If $m_1$ and $m_2$ are the slopes of the pair of lines given by $ax^2+2hxy+by^2=0$, then the value of $m_1+m_2$ is :

For $ax^2+2hxy+by^2=0$, the sum of the slopes is $m_1+m_2=-\dfrac{2h}{b}$. Dividing the equation $ax^2+2hxy+by^2=0$ by $x^2$ and writing $m=\dfrac{y}{x}$ (the slope of each line through the origin) gives the quadratic in $m$: $bm^2+2hm+a=0.$ By the relation between the roots and coefficients of a quadratic: $m_1+m_2=-\frac{2h}{b},\qquad m_1 m_2=\frac{a}{b}.$ …