If the determinant `|[a,b,2aalpha+3b],[b,c,2balpha+3c],[2aalpha+3b,2balpha+3c,0]|=0` then (a)`a , b , c` are in H.P. `alpha` is root of `4a x^2+12 b x+9c=0` or `a , b , c` are in G.P. `a , b , c ,` are in G.P. only `a , b ,c` are in A.P.

Explanation

To solve the determinant problem step by step, we start with the given determinant:

\[
D = \begin{vmatrix}
a & b & 2a\alpha + 3b \\
b & c & 2b\alpha + 3c \\
2a\alpha + 3b & 2b\alpha + 3c & 0
\end{vmatrix}
...