If a `DeltaABC, a,b,` A are given and `b_(1), b_(2)` are two values of the third sides b such that `b_(2)=2b_(1).` Then, sin A is equal to

Options

A. `sqrt((9a^(2)-c^(2))/(8a^(2)))`
B. `sqrt((9a^(2)-c^(2))/(8c^(2)))`
C. `sqrt((9a^(2)+c^(2))/(8a^(2)))`
D. none of these

Answer

B

Explanation

We have,
`cosA=(b^(2)+c^(2)-a^(2))/(2bc)impliesb^(2)-2bc cosA+(c^(2)-a^(2))=0`
It is given that `b_(1)` and `b_(2)` are roots of this equation.
`therefore b_(1)+b_(2)=2c cos A and b_(1)b_(2)=c^(2)-a^(2)`
`implies 3 b_(1)=2c cosA and, 2b_(1)^(2)=c^(2)-a^(2) " "[because b_(2)=2b_(1)"given"]`
`implies((2c)/(3)cosA)^(2)=c^(2)-a^(2)`
`implies 8c^(2)(1-sin^(2)A)=9c^(2)-9a^(2)impliessinA=sqrt((9a^(2)-c^(2))/(8c^(2)))`