Question

If the family of lines $x+\left(a+2b\right)+y\left(a+3b\right)=a+b$ passes through the point for all values of $a$ and $b$, then the coordinates of the point are:

• A. $(2,1)$
• B. $(2,-1)$
• C. $(-2,1)$
• D. None of these

Answer 1

The correct option is B $(2,-1)$

$x(a+2b)+y(a+3b)=a+b$

$a(x+y-1)+b(2x+3y-1)=0$

The family of lines will pass through intersection of

$x+y-1=0$ & $2x+3y-1=0$

Now, $x+y-1=0$ $\Rightarrow 3x+3y-3=0$

$2x+3y-1=0$

$\underset{–}{--+}$

$x-2=0$ $\Rightarrow x=2$

$\Rightarrow$ $y=-1$

$\therefore$ $(2,-1)$ is required point [B]