Question

Find the condition that one of the lines given by the equation $a{x}^2+2hxy+b{y}^2=0$ coincides with one of those given by $a^{\prime }{x}^2+2h^{\prime }xy+b^{\prime }{y}^2=0$.

Answer 1

If the line $y=mx$ be common to both the pairs then
putting $mx$ for $y$ in the two equations, we get
$a{x}^2+2hm{x}^2+b{x}^2{m}^2=0$
or $b{m}^2+2hm+a=0$
$d{x}^2+2hm{x}^2+b{x}^2{m}^2=0$
or $b{m}^2+2hm+d=0$
$\therefore \frac{m^2}{(dh-ah)}=\frac{2m}{ab-db}=\frac{1}{bh-bh}$ by $\left(1\right),\left(2\right)$
$\therefore m=\frac{2(dh-ah)}{(ab-db)}$
and $m=\frac{(ab-db)}{2(bh-bh)}$.
$\therefore \frac{2(dh-ah)}{(ab-db)}=\frac{(ab-db)}{2(bh-bh)}$
or $(ab-db{)}^2-4(dh-ah\left)\right(bh-bh)=0$
is the required condition.