Question

find the condition that the lines ax+hy+g=0, hx+by+f=0, gx+fy+c=0 are to be concurrent

Answer 1

Answer:

The required condition is

$abc+2fgh-af^2-bg^2-ch^2=0$

Step-by-step explanation:

since the lines ax+hy+g=0, hx+by+f=0, gx+fy+c=0 are concurrent,

$\left|\begin{array}{ccc}a&h&g\\h&b&f\\g&f&c\end{array}\right|=0$

Expanding along first row, we get

$a(bc-f^2)-h(hc-fg)+g(fh-bg)=0$

$abc-af^2-ch^2+fgh+fgh-bg^2=0$

$abc+2fgh-af^2-bg^2-ch^2=0$

This is the required condition.