Question

If an equation $hxy+gx+fy+c=0$ represents a pair of lines, then

• A. $fg=ch$
• B. $gh=cf$
• C. $fh=cg$
• D. $hf=-cg$

Answer 1

The correct option is A $fg=ch$

The given equation $hxy+gx+fy+c=0$ will represent pair of straight line,

If $\left|\begin{array}{ccc}0& \frac{h}{2}& \frac{-g}{2}\\ \frac{h}{2}& 0& \frac{-f}{2}\\ \frac{-g}{2}& \frac{-f}{2}& c\end{array}\right|=0$

Expand along $R_1$

We get,

$0.[0-\frac{f^2}{2}]-\frac{h}{2}[\frac{hc}{2}-\frac{gf}{4}]-\frac{g}{2}[\frac{-fh}{4}-0]=0$

$0-\frac{h}{2}[\frac{hc}{2}-\frac{gf}{4}]-\frac{g}{2}\left[\frac{-fh}{4}\right]=0$

$-\frac{h}{2}[\frac{hc}{2}-\frac{gf}{4}]=\frac{g}{2}\left[\frac{-fh}{4}\right]$

$-\frac{h^{2}c}{4}+\frac{ghf}{8}=\frac{-gfh}{8}$

$-\frac{h^{2}c}{4}=\frac{-gfh}{8}-\frac{gfh}{8}$

$-\frac{h^{2}c}{4}=\frac{-gfh}{4}$

$hc=gf$

$fg=ch$