Question

If the line ${ay}^2+bxy+ex+dy=0$ represents a pair of lines, then

• A. $bd-ae=0$ or $e=0$
• B. $be-ad=0$ or $e=0$
• C. $ad-eb=0$ $e=0$
• D. $ad+be=0$ $e=0$

Answer 1

The correct option is A $bd-ae=0$ or $e=0$

The equation $S=A{x}^2+2Hxy+B{y}^2+2Gx+2Fy+C=0$ represents a pair of straight lines if the following conditions hold :

$\Delta =ABC+2FGH-A{F}^2-B{G}^2-C{H}^2=0$

$H^2\ge AB,G^2\ge AC,F^2\ge BC$

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

$\Delta =0+\frac{bde}{4}-0-\frac{a{e}^2}{4}-0=0$

$\Rightarrow bde-a{e}^2=0$

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