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`
`a_(2)x+b_(2)y +1=0`
` a_(3) x+b_(3)y+1=0`
if these lines are concurrent we must have3 `|underset(a_(3)" "b_(3)" "1)underset(a_(2)" "b_(2)" "1)(a_(1)" "b_(1)" "1)|=0`
Which is the condition of colinearity of three points `(a_(1) b_(1)), (a_(2),b_(2)), " and " (a_(3),b_(3))`
Hence if the given lies are concurrent the given points are collinear.