Question

If the lines `a_1x+b_1y+1=0, a_2x+b_2y+1=0 a n d a_3x+b_3y+1=0` are concurrent, show that the points `(a_1, b_1), (a_2, b_2)a n d (a_3, b_3)` are collinear.

Answer 1

The given lines are
`a_(1)x+b_(1)y+1=0 " "(1)`
`a_(2)x+b_(2)y+1=0 " "(2)`
`"and " a_(3)x+b_(3)y+1=0 " "(3)`
If these lines are concurrent, we must have
`|{:(a_(1),b_(1),1),(a_(2), b_(2), 1),(a_(3),b_(3),1):}| =0`
which is the condition of the collinearity of these points `(a_(1), b_(1)),(a_(2), b_(2)) " and " (a_(3), b_(3)).`
Hene, if the given lines are concurrent, the given points are collinear.