Question

If the pair of straight lines $xy-x-y+1=0$ and the line $ax+2y-3=0$ are concurrent , then $a=?$

• A. $-1$
• B. $0$
• C. $3$
• D. $1$

Answer 1

The correct option is D

$1$

Explanation for correct option:

Given equation is $xy-x-y+1=0$

$$\begin{gathered} \Rightarrow x\left(y-1\right)-\left(y-1\right)=0 \\ \Rightarrow \left(x-1\right)\left(y-1\right)=0 \end{gathered}$$

$\Rightarrow x-1=0$or $y-1=0$

Let $L_1:x-1=0$

$L_2:y-1=0$

$L_3=ax+2y-3=0$ (given)

Given $L_1,L_2,L_3$ are concurrent

$$\begin{gathered} \Rightarrow \left|\begin{array}{ccc}a& 2& -3\\ 1& 0& -1\\ 0& 1& -1\end{array}\right|=0 \\ \Rightarrow a\left(0-\left(-1\right)\right)-2\left(-1-0\right)-3\left(1-0\right)=0 \\ \Rightarrow a+2-3=0 \\ \Rightarrow a=1 \end{gathered}$$

Hence, option (D) $1$, is the correct answer.