if `x gt m,y gt n,z gt r (x,y,zgt 0)` such that `|{:(x,,n,,r),(m,,y,,r),(m,,n,,z):}|=0`
the value `(xyz)/((x-m)(y-n)(z-r))` is

Options

A. 27
B. `(8)/(27)`
C. `(64)/(27)`
D. None of these

Answer

B

Explanation

`|{:(x,,n,,r),(m,,y,,r),(m,,n,,z):}|=0`
Applying `R_(1) to R_(1)-R_(2) " and " R_(2) to R_(2)-R_(3)` we get
`|{:(x-m,,n-y,,0),(0,,y-n,,r-z),(m,,n,,z):}| =0`
`rArr (x-m)(y-n)(n-y)(r-z)m-n(r-z)(x-m) =0`
Dividing by `(x-m)(y-n)(z-r)` we have
`(z)/(z-r)+(m)/(x-m) +(n)/(y-n) =0`
`" or " (z)/(z-r) +(m)/(x-m) +1 +(m)/(y-n) +1=2`
` " or " (z)/(z-r) +(x)/(x-m) +(y)/(y-n) =2`
`" or " (m)/(x-m)+(n)/(y-n)+(r)/(z-r)=-1`
Now `,A.M ge G.M.`
`rArr ((z)/(z-r)+(x)/(x-m)+(y)/(y-n))/(3)ge ((z)/((z-r))(x)/((x-m))(y)/((y-n)))^(1//3)`
`" or " (z)/(z-r)(x)/(x-m) (y)/(y-n) le (8)/(27)`