Example
Q
Find the condition that the lines ax + hy + g = 0, hx + by + f = 0, gx + fy + c = 0 to be concurrent.
Sol:
Let P(α, β) be the point of concurrence
aα + hβ + g = 0 ------ (1)
hα + bβ + f = 0 ------ (2)
gα + fβ + c = 0 ------ (3)
From (1) and (2) we get
⇒ g(hf – bg) + f(gh – af) + c(ab – h
2
) = 0
⇒ fgh – bg
2
+ fgh – af
2
+ abc – ch
2
= 0
⇒ abc + 2fgh – af
2
– bg
2
– ch
2
= 0
∴ The condition is abc + 2fgh – af
2
– bg
2
– ch
2
= 0.
i.e.,