If a,b,c are in A.P and `a^2,b^2,c^2` are in H.P then which is of the following is /are possible ?

Options

A. `ax^2 +bx+c =0`
B. `ax^2bx+c=0`
C. `a,b-c/2` form a G.P
D. `a-b, c/2` from a G.P

Answer

A, C

Explanation

Given that a,b,c are in A.P.
`rArr2b=a+c` …(1)
and `a^(2),b^(2),c^(2)` are in H.P.
`rArr1/b^(2)-1/a^(2)=1/c^(2)-1/b^(2)`
`rArr((a-b)(a+b))/(b^(2)a^(2))=((b-c)(b+c))/(b^(2)c^(2))`
`rArr ac^(2)+bc^(2)=a^(2)b+a^(2)c` [`because a-b=b-c`]
`rArrac(c-a)+b(c-a)(c+a)=0`
`rArr(c-a)(ab+bc+ca)=0`
`rArrc-a=0` or ab+bc+ca=0
For c=a, from (1),a=b=c
For (a+c)b+ca=0, from (1),
`2b^(2)+ca=0`
`rArrb^(2)=a((-c)/2)`
This implies that a,b,-c/2 are in G.P.