Question

If $a,b,c$ are in arithmetic progression, then show that the equation $ax+by+c=0$ represents a family of concurrent lines and find the pint of concurrency.

Answer 1

If a, b, c are in A.P.

$\Rightarrow b-a=c-b$

$\Rightarrow 2b=a+c$

$\Rightarrow ar+\left(\frac{x+c}{2}\right)y+c=0$

$\Rightarrow a(x+\frac{y}{2})+c(\frac{y}{2}+1)=0$

$\Rightarrow$ This represents family of lines passing through point of intersection of $x+\frac{y}{2}=0,\frac{y}{2}=-1$

P.O.I $\Rightarrow y=-2$ and $x=-\frac{y}{2}=\frac{2}{2}=1$

$\Rightarrow$ P.O.I$=(1,-2)$.