Question

If $\Delta =\left|\begin{array}{ccc}x+1& x+2& x+a\\ x+2& x+3& x+b\\ x+3& x+4& x+c\end{array}\right|=0$, then
the family of lines $ax+by+c=0$ passes through

• A. $(1,-1)$
• B. $(1,-2)$
• C. $(2,-3)$
• D. $(0,0)$

Answer 1

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

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

Applying $C_2\to C_2-C_1,C_3\to C_3-C_1$

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

Applying $R_2\to R_2-R_1,R_3\to R_3-R_1$

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

Expanding along $C_2$
$1(c-a-2)-2(b-a-1)=0\Rightarrow a-2b+c=0$
From options for point $(1,-2)$ lies on the line $ax+by+c=0$