MCQ
The determinant $\mathrm{D}=\left|\begin{array}{ccc}a & b & a+b \\ b & c & b+c \\ a+b & b+c & 0\end{array}\right|=0$, if
- Aa, b, c are in A.P.
- ✓a, b, c are in G.P.
- Ca, b, c are in H.P.
- D$\alpha$ is a root of $a x^2+2 b x+c=0$V
Hint:
Applying $R_3 \rightarrow R_3-\left(R_1+R_2\right)$, we get
$\left|\begin{array}{ccc}a & b & a+b \\ b & c & b+c \\ 0 & 0 & -(a+2 b+c)\end{array}\right|=0$
$\begin{aligned} & \therefore a[-c(a+2 b+c)-0]-b[-b(a+2 b+c)-0]+(a+b)(0-0)=0 \\ & \therefore\left(-a c+b^2\right)(a+2 b+c)=0 \\ & \therefore-a c+b^2=0 \text { or } a+2 b+c=0 \\ & \therefore b^2=a c \\ & \therefore a, b, c \text { are in G.P. }\end{aligned}$
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