If the lines `x+a y+a=0,\ b x+y+b=0\ a n d\ c x+c y+1=0` are concurrent, then write the value of `2abc-ab-bc-ca dot`

Explanation

The given lines are `x + ay + a = 0` … (1)

`bx + y + b = 0 `… (2)

`cx + cy + 1 = 0` … (3)

It is given that the lines (1), (2) and (3) are concurrent.

so,`[[1,a,a],[b,1,b],[c,c,1]]=0`

`=>(1 – bc) – a(b – bc) + a(bc – c) = 0`

`=>1 – bc – ab + abc + abc – ac = 0`

`=>2abc – ab – bc – ca = -1` Hence, the value of` 2abc − ab − bc − ca` is − 1`