Question

If $3a+2b+4c=0$, then show that the equation $ax+by+c=0$ represents a family of concurrent straight line and find the point of concurrency.

Answer 1

$b=\frac{-4c-3a}{2}$

$\Rightarrow ax+\left(\frac{-4c-3a}{2}\right)y+c=0$

$\Rightarrow a(x-\frac{3y}{2})+c(1-\frac{4y}{2})=0$

$\Rightarrow$ It represents family of lines passing through point of intersection of $L_1,L_2$

P.O.I $\frac{4y}{2}=1$

$\Rightarrow y=\frac{1}{2}$;

$x-\frac{3y}{2}=0$

$\Rightarrow x=\frac{3}{2}\times \frac{1}{2}$

$\Rightarrow x=\frac{3}{4}$

$\Rightarrow$ P.O.I. $=\frac{3}{4},\frac{1}{2}$.