Question

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.

Answer 1

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.

$$\therefore \left|\begin{array}{ccc}1& a& a\\ b& 1& b\\ c& c& 1\end{array}\right|=0$$

$$\Rightarrow (1-bc)-a(b-bc)+a(bc-c)=0$$

$$\Rightarrow 1-bc-ab+abc+abc-ac=0$$

$$\Rightarrow 2abc-ab-bc-ca=-1$$

Hence, the value of 2abc − ab − bc − ca is −1