Question

If the three distinct lines $x+2ay+a=0,$ $x+3by+b=0$ and $x+4ay+a=0$ are concurrent, then the point $(a,b)$ lies on

• A. a circle
• B. a parabola
• C. an ellipse
• D. a straight line

Answer 1

The correct option is D a straight line

$x+2ay+a=0\dots \left(1\right)$
$x+3by+b=0\dots \left(2\right)$
$x+4ay+a=0\dots \left(3\right)$
Solving $\left(1\right)$ and $\left(3\right),$
$y=0,x=-a$
As given three lines are concurrent,
so, $(-a,0)$ will satisfy the equation $\left(2\right)$
$-a+b=0$
$\Rightarrow a=b$
$\therefore$ Locus of $(a,b)$ is $x=y$
which is a straight line.