Question

If the equation $a{x}^2+b{y}^2+cx+cy=0$, represents a pair of straight lines, then

• A. $a(b+c)=0$
• B. $b(c+a)=0$
• C. $c(a+b)=0$
• D. $a+b+c=0$

Answer 1

The correct option is C

$c(a+b)=0$

Explanation for correct option

Step 1: Comparing with the general equation of pair of straight line

Given equation is $a{x}^2+b{y}^2+cx+cy=0$

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

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

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

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

$h{\text{'}}^2>a\text{'}b\text{'}$

$g{\text{'}}^2>a\text{'}c\text{'}$

From the first condition we have,

$$\begin{gathered} a\text{'}b\text{'}c\text{'}+2f\text{'}g\text{'}h\text{'}-a\text{'}f{\text{'}}^2-b\text{'}g{\text{'}}^2-c\text{'}h{\text{'}}^2=0 \\ \Rightarrow 0+0-\frac{a{c}^2}{2}-\frac{b{c}^2}{2}-0=0 \\ \Rightarrow a{c}^2+b{c}^2=0 \\ \Rightarrow c^2\left(a+b\right)=0 \\ \Rightarrow c\left(a+b\right)=0 \end{gathered}$$

Hence, Option (C) is correct i.e. $c(a+b)=0$