If a,b,c are in A.P. and `a^2,b^2,c^2` are in H.P. then which of the following could and true (A) `-a/2, b, c are in G.P.` (B) `a=b=c` (C) `a^3,b^3,c^3` are in G.P. (D) none of these

Options

A. `-(a)/(2),b,c` are in GP
B. `a=b=c`
C. `a^(2),b^(2),c^(2)` are in GP
D. None of these

Answer

A, B

Explanation

`:.a,b,c` are in AP `" " Implies b=(a+c)/(2)" " "……..(i)"`
and `a^(2),b^(2),c^(2)` arein HP.
` impliesb^(2)=(2a^(2)c^(2))/(a^(2)+c^(2))" " "......(ii)"`
`implies b^(2){a^(2)+c^(2)}=2a^(2)c^(2)`
`implies b^(2){(a+c)^(2)-2ac}=2a^(2)c^(2)" "[" from Eq.(i) "]`
`implies 2b^(4)-ac(b^(2))-a^(2)c^(2)=0`
`implies (b^(2)-ac)(2b^(2)+ac)=0`
If `b^(2)-ac=0`
a,b,c are in GP.
But given a,b,c are in AP.
`:.a=b=c`
and if `2b^(2)+ac=0` ltbrlt then `(-a)/(2),b,c` are in GP.