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Question

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


A
0
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B
1
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C
1
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D
ab
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Solution

The correct option is A $$0$$
Given $$a,b,c$$ are in A.P.
Performing  $$ C_{3}\rightarrow C_{3}-xC_{1}-yC_{2}$$
                     
$$=\begin{vmatrix}a & b & ax+by\\ b & c & bx+cy\\ 0 & 0 & -ax^{2} -2bxy-cy^{2}\end{vmatrix}$$
$$=-\left ( ax^{2}+2bxy+cy^{2} \right )\left ( ac-b^{2} \right )$$
$$= 0    \left \{ \because b^{2}=ac \right \}$$

Mathematics

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