If the lines x + ay + a = 0, bx + y + b = 0 and cx + cy + 1 = 0 are concurrent then write the value of 2abc – ab – bc – ca.

Given:

Lines x + ay + a = 0, bx + y + b = 0 and cx + cy + 1 = 0

To find:

The value of 2abc – ab – bc – ca.

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.

∴

⇒ (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