Question

If the lines ax + y + 1 = 0, x + by + 1 = 0 and x + y + c = 0 (a, b, c being distinct and different from 1 ) are concurrent, then value of $\frac{1}{1-a}+\frac{1}{1-b}+\frac{1}{1-c}$ 

a) -1

b) 0

c) 1

d) 2

Answer 1

The correct answer is option c) 1

if the given lines are concurrent  , then $\overline{)\begin{array}{ccc}a& 1& 1\\ 1& b& 1\\ 1& 1& c\end{array}}=0$  

Applying { $C_2\to C_2-C_{1}and{C}_3\to C_3-C_1$} 

$\overline{)\begin{array}{ccc}a& 1-a& 1-a\\ 1& b-1& 0\\ 1& 0& c-1\end{array}}=0$

= a(b-1)(c-1) -(c-1)(1-a)-(b-1)(1-a)=0

=$\frac{a}{1-a}+\frac{1}{1-b}+\frac{1}{1-c}$ 

(dividing by (1-a )(1-b)(1-c)) 

$=\frac{1}{1-a}+\frac{1}{1-b}+\frac{1}{1-c}=1$