Question

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

• A. $fh=cg$
• B. $fg=ch$
• C. $h^2=gf$
• D. $fgh=c$

Answer 1

The correct option is B

$fg=ch$

Explanation for correct option

Step 1: Solve for the coefficients

Given equation is $hxy+gx+fy+c=0$

The general equation of the pair of straight line is $a\text{'}{x}^2+2h\text{'}xy+b\text{'}{y}^2+2g\text{'}x+2f\text{'}y+c\text{'}=0$

Now comparing with the general equation of the straight line we get

$a\text{'}=0,h\text{'}=\frac{h}{2},b\text{'}=0,g\text{'}=\frac{g}{2},f\text{'}=\frac{f}{2},c\text{'}=c$

Step 2: Solve for the condition

The necessary and sufficient condition for general equation $a\text{'}{x}^2+2h\text{'}xy+b\text{'}{y}^2+2g\text{'}x+2f\text{'}y+c\text{'}=0$ represent a pair of straight line is

1. $a\text{'}b\text{'}c\text{'}+2f\text{'}g\text{'}h\text{'}-a\text{'}f{\text{'}}^2-b\text{'}g{\text{'}}^2-c\text{'}h{\text{'}}^2=0$
2. $h{\text{'}}^2>a\text{'}b\text{'}$
3. $g{\text{'}}^2>a\text{'}c\text{'}$

From the first condition we have,

$$\begin{gathered} a\text{'}b\text{'}c\text{'}+2f\text{'}g\text{'}h\text{'}-a\text{'}f{\text{'}}^2-b\text{'}g{\text{'}}^2-c\text{'}h{\text{'}}^2=0 \\ \Rightarrow 0+\frac{fgh}{4}-0-0-\frac{c{h}^2}{4}-0=0 \\ \Rightarrow fgh-c{h}^2=0 \\ \Rightarrow fgh=c{h}^2 \\ \Rightarrow fg=ch \end{gathered}$$

Hence, Option (B) is correct i.e. $fg=ch$