Question

Find the condition that the equation
$a{y}^2+bxy+ex+dy=0$ may represent of lines.

Answer 1

Comparing the equation $a{y}^2+bxy+ex+dy=0$ with $A{x}^2+2Hxy+b{y}^2+2Gx+2Fy+C=0$, we get

$A=0,H=\frac{b}{2},B=a,G=\frac{e}{2},F=\frac{d}{2},C=0$

The given equation represents a pair of lions,

if $\left|\begin{array}{ccc}A& H& G\\ H& B& F\\ G& F& C\end{array}\right|=0$

i.e. if $\left|\begin{array}{ccc}0& \frac{b}{2}& \frac{e}{2}\\ \frac{b}{2}& a& \frac{d}{2}\\ \frac{e}{2}& \frac{d}{2}& 0\end{array}\right|=0$

i.e if $0-\frac{b}{2}(0-\frac{ed}{4})+\frac{e}{2}(\frac{bd}{4}-\frac{ae}{2})=0$

i.e. if $\frac{bed}{8}+\frac{bed}{8}-\frac{a{e}^2}{4}=0$

i.e. if $bed-a{e}^2=0$

i.e. if $e=0$ or $bd-ae=0$

i.e. if $e=0$ or $bd=ae$

This is the required condition.