Question

If $\left|\begin{array}{ccc}a& b& a+b\\ b& c& b+c\\ a+b& b+c& 0\end{array}\right|=0$, then $a,b,c$ are in

• A. A.P.
• B. G.P.
• C. H.P.
• D. None of these

Answer 1

The correct option is B G.P.

$\left|\begin{array}{ccc}a& b& a+b\\ b& c& b+c\\ a+b& b+c& 0\end{array}\right|=0$

Applying $R_3\to R_3-(R_1+R_2)$ we have :-

$⟹$ $\left|\begin{array}{ccc}a& b& a+b\\ b& c& b+c\\ 0& 0& -(a+2b+c)\end{array}\right|=0$

Applying $R_1\leftrightarrow R_3$ we have :-

$⟹$ $(-1)\ast$$\left|\begin{array}{ccc}0& 0& -(a+2b+c)\\ b& c& b+c\\ a& b& a+b\end{array}\right|=0$

Now expanding along $R_3$ we have :-

$⟹$$[0+0-(a+2b+c\left)\right(b^2-ac\left)\right]$ $=0$

$⟹$ $-(a+2b+c)(b^2-ac)=0$

So either $(a+2b+c)=0$ or $(b^2-a\ast c)=0$

Case-1 when $(a+2b+c)=0$

$⟹$ $a+2b=-c$

So no relation in this case

Case-2 when $(b^2-ac)=0$

$⟹$ $b^2=ac$

Hence, $a,b,c$ are in G.P.