Question

If $\left|\begin{array}{ccc}x+1& x+2& x+a\\ x+2& x+3& x+b\\ x+3& x+4& x+c\end{array}\right|=0$ then all lines represented by $ax+by+c=0$ pass through a fixed a point. The fixed point is

• A. $(1,-2)$
• B. $(1,2)$
• C. $(-1,-2)$
• D. $(-1,2)$

Answer 1

The correct option is A $(1,-2)$

We have,

$\left|\begin{array}{ccc}x+1& x+2& x+a\\ x+2& x+3& x+b\\ x+3& x+4& x+c\end{array}\right|=0$

$C_2\to C_2-C_1\left|\begin{array}{ccc}x+1& 1& x+a\\ x+2& 1& x+b\\ x+3& 1& x+c\end{array}\right|=0$

$R_2\to R_2-R_1,R_3\to R_3-R_2\left|\begin{array}{ccc}x+1& 1& x+a\\ 1& 0& b-a\\ 1& 0& c-b\end{array}\right|=0-\left[\left(c-b\right)-\left(b-a\right)\right]=0c+a=2ba-2b+c=0ax+by+c=0$

$Oncomparingx=1y=-2fixedprovethat:\left(1,-2\right)$

Option $A$ is correct answer.