Question

What relations must hold between the coordinates of the equations
(i) $a{x}^2+b{y}^2+cx+cy=0$
and
(ii) $a{y}^2+bxy+dy+ex=0$
so that each of them may represent a pair of straight lines ?

Answer 1

The general equation of second degree $a{x}^2+2hxy+{by}^2+2gx+2fy+c=0$

Represents a pair of straight lines if $\Delta =abc+2fgh-a{f}^2-b{g}^2-c{h}^2=0$

For $\left(i\right)$

we have $a=a,b=b,h=0,g=\frac{c}{2},f=\frac{c}{2},c=0$

$\Delta =(a\times b\times 0)+(2\times \frac{c}{2}\times \frac{c}{2}\times 0)-(a\times \frac{c}{2}\times \frac{c}{2})-(b\times \frac{c}{2}\times \frac{c}{2})-(0\times 0\times 0)=00+0-a.\frac{c^2}{2}-b.\frac{c^2}{2}-0=0a{c}^2+b{c}^2=0c^2(a+b)=0$

$\Rightarrow c=0$ or $a+b=0$

For $\left(ii\right)$

we have $a=0,b=a,h=\frac{b}{2},g=\frac{e}{2},f=\frac{d}{2},c=0$

$\Delta =(0\times a\times 0)+(2\times \frac{d}{2}\times \frac{e}{2}\times \frac{b}{2})-(0\times \frac{d}{2}\times \frac{d}{2})-(a\times \frac{e}{2}\times \frac{e}{2})-(0\times \frac{b}{2}\times \frac{b}{2})=0\Delta =0+\frac{bed}{4}-0-\frac{a{e}^2}{4}-0=0bed-a{e}^2=0e(bd-ae)=0$

$\Rightarrow e=0$ or $ae=bd$