First slide

Equation of line in Straight lines

Question

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

Moderate

A

-1
B

0
C

1
D

none of these

Solution

$\begin{array}{l} \frac{x}{a} + y + 1 = 0, x + \frac{y}{b} + 1 = 0, \\ x + y + \frac{1}{c} = 0 \\ x = \frac{c - 1}{a - 1} \cdot \frac{a}{c}, y = \frac{c - 1}{b - 1} \cdot \frac{b}{c} \\ x + y + \frac{1}{c} = 0 \Rightarrow \frac{a}{a - 1} + \frac{b}{b - 1} + \frac{1}{c - 1} = 0 \\ \Rightarrow \frac{a}{a - 1} + \frac{b}{b - 1} + \frac{c}{c - 1} = 1 \end{array}$

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