If a,b,c are three distinct numbers in G.P., b,c,a are in A.P and a,bc, abc, in H.P then the possible value of b is

Options

A. `3+4sqrt(2)`
B. `3-4 sqrt(2)`
C. `4+ 3 sqrt(2)`
D. `4-3sqrt(2)`

Answer

C, D

Explanation

Let `a=A/r,b=A and C=Ar`
Now b,c,a are in A.P..
`therefore2c=a+b`
`rArr2Ar=A/r+A`
`rArr2r^(2)-r-1=0`
`rArrr=1 or r=-1/2`
Sine a,b, are distinctt, `r=-1/2`
`thereforea=-2A,b=A,c=-A/2`
a,bc,abc are in H.P.
`therefore 1/a,1/(bc),1/(abc)` are in A.P.
`rArr 2/(bc)=1/(abc)+1/a`
`rArr-4/A^(2=1/A^(3)-1/(2A)`
`rArr-4/(A^(2))=(2-A^(2))/(2A^(3))`
`rArr-8A=2-A^(2)`
`rArrA^(2)-8A-2=0`
`rArrA=pm3sqrt2`