For any three positive real numbers `a , b` and `c ,9(25 a^2+b^2)+25(c^2-3a c)=15 b(3a+c)dot` Then : `a ,b` and`c` are in`AdotPdot` (2) `a ,b` and `c` are in `GdotPdot` `b ,c` and `a` are in `GdotPdot` (4) `b , c` and `a` are in `AdotPdot`

Options

A. a,b and c are in GP
B. b,c and a are in GP
C. b,c and a are in AP
D. a,b and c are in AP

Answer

C

Explanation

`(15a)^(2)+(3b)^(2)+(5c)^(2)-45ab-15bc-75ac=0`
`implies (1)/(2){(15a-3b)^(2)+(3b+5c)^(2)+(5c-15a)^(2)}=0`
`implies (15a-3b)^(2)+(3b+5c)^(2)+(5c-15a)^(2)=0`
or `15a-3b=0,3b-5c=0,5c-15a=0`
`:. b=5a,c=3a`
`implies 5a,3a,a` are in AP i.e b,c,a are in AP.