Question

If a,b,c are in G.P. then the value of $\left|\begin{array}{ccc}a& b& ax+by\\ b& c& bx+cy\\ ax+by& bx+cy& 0\end{array}\right|$ is

• A. $0$
• B. $1$
• C. $-1$
• D. $ab$

Answer 1

The correct option is A $0$

Given $a,b,c$ are in A.P.

Performing $C_3\to C_3-x{C}_1-y{C}_2$

$=\left|\begin{array}{ccc}a& b& ax+by\\ b& c& bx+cy\\ 0& 0& -a{x}^2-2bxy-c{y}^2\end{array}\right|$
$=-(a{x}^2+2bxy+c{y}^2)(ac-b^2)$
$=0\{\because b^2=ac\}$