If `a,b,c` are in AP and `A,B,C` are in G.P. (Common ratio `!=1`). Then which of the following is/are correct

Options

A. `A/a,B/b,C/c` are in HP if common of GP is `c//a`
B. `a/A,b/B,c/C` are in HP if common ratio of GP is equal to common difference of AP
C. `(A^(2))/a,(B^(2))/b,(C^(2))/c` are in HP if common ratio of GP is `sqrt(c/a)`
D. `a/(A^(2)),b/(B^(2)),c/(C^(2))` are in HP if common ratio of GP is equal to square root of common difference of AP.

Answer

A, B, D

Explanation

`A/a,B/b,C/c HP`
`=(2b)/B=a/A+c/C`
`implies2bB=aC+cA`
`impliesaB+cB=aC+cA`
`impliesa[B-C]=c[A-B]`
so `r=c/a`
`(A^(2))/a,(B^(2))/b,(C^(2))/c` are in `HPimpliesr^(2)=c/a`