Question

The equations of the lines represented by the equation $a{x}^2+\left(a+b\right)xy+b{y}^2+x+y=0$ are

• A. $ax+by+1=0,x+y=0$
• B. $ax+by-1=0,x+y=0$
• C. $ax+by+1=0,x-y=0$
• D. None of these

Answer 1

The correct option is A

$ax+by+1=0,x+y=0$

Calculate the equations of lines:

The given equation is $a{x}^2+\left(a+b\right)xy+b{y}^2+x+y=0$.

$$\begin{gathered} a{x}^2+\left(a+b\right)xy+b{y}^2+x+y=0 \\ ax\left(x+y\right)+by\left(y+x\right)+x+y=0 \\ \left(ax+by+1\right)\left(x+y\right)=0 \\ ax+by+1=0,x+y=0 \end{gathered}$$

Hence, the equations of the lines are $ax+by+1=0,x+y=0$ and therefore option $\left(A\right)$ is correct.