If the determinant `|[a ,b,2a alpha+3b],[b ,c, 2b alpha+3c],[2a alpha+3b,2b alpha+3c,0]|=0 then `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

If the determinant `|[a ,b,2a alpha+3b],[b ,c, 2b alpha+3c],[2a alpha+3b,2b alpha+3c,0]|=0`


Applying `C_3→C_3−2αC_1−3C_2,`

we get `|[a ,b,0],[b, c, 0],[2a alpha+3b,2b alpha+3c,-2alpha(2a alpha+3b)-3(2b alpha+3c)]|=0`

⇒`−(ac−b^2){2α(2aα+3b)+3(2bα+3c)}=0`

⇒`(b^2−ac){4aα^2+12bα+9c}=0 `

⇒`b^2−ac=0,4aα^2+12bα+9c=0 `

⇒a,b,c are in G.P. and x=α is a root of `4ax^2+12bx+9c=0`