Question

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

Hint: In this question, we are to find the equation of the lines that are represented by the given equation. Since the given equation represents two lines, we can simplify the given equation into factors. Those factors are the required equations for the lines.



Formula Used:The combined equation of the pair of straight lines is
$H\equiv a{{x}^{2}}+2hxy+b{{y}^{2}}=0$
This is called a homogenous equation of the second degree in $x$ and $y$
And
$S\equiv a{{x}^{2}}+2hxy+b{{y}^{2}}+2gx+2fy+c=0$
This is called a general equation of the second degree in $x$ and $y$.
If $S\equiv a{{x}^{2}}+2hxy+b{{y}^{2}}+2gx+2fy+c=0$ represents a pair of lines, then
i) $abc+2fgh-a{{f}^{2}}-b{{g}^{2}}-c{{h}^{2}}=0$ and
ii) ${{h}^{2}}\ge ab,{{g}^{2}}\ge ac,{{f}^{2}}\ge bc$



Complete step by step solution:Given equation is
$a{{x}^{2}}+(a+b)xy+b{{y}^{2}}+x+y=0$
Since it represents two lines, simplifying it into factors as follows:
$\begin{align}
  & a{{x}^{2}}+(a+b)xy+b{{y}^{2}}+x+y=0 \\
 & \Rightarrow a{{x}^{2}}+axy+bxy+b{{y}^{2}}+x+y=0 \\
 & \Rightarrow ax(x+y)+by(x+y)+(x+y)=0 \\
 & \Rightarrow (x+y)(ax+by+1)=0 \\
\end{align}$
We got two factors for the given equation. They are
$x+y=0$
And
$ax+by+1=0$
Therefore, these factors represent equation of lines. So, these are pair of the line that are represented by the equation $a{{x}^{2}}+(a+b)xy+b{{y}^{2}}+x+y=0$.



Option ‘A’ is correct



Note: Here, it is given that the given equation represents two lines. So, by simplifying it we get the required equation. Here simplification is nothing but factorizing the given equation into factors. Thus, the factors obtained from the given equation are the required lines. For some other questions, we need to use the general form of the pair of lines equation.