Let a,b,c and d are real numbers in GP. Suppose u,v,w satisfy the system of equations `u+2y+3w=6,4u+5y+6w=12` and `6u+9v=4`. Further consider the expressions
`f(x)=(1/u+1/v+1/w)x^(2)+[(b-c)^(2)+(c-a)^(2)+(x-b)^(2)]`
`x+u+v+w=0` and `g(x)=20x^(2)+10(a-d)^(2)x-9=0`
`(u+v+w)` is equal to

Options

A. `a-d`
B. `(a-d)^(2)`
C. `a^(2)-d^(2)`
D. `(a+d)^(2)`

Explanation

Let `b=ar,c=ar^(2)` and `d=ar^(3)`
Now `(b-c)^(2)+(c-a)^(2)+(d-b)^(2)`
`=(ar-ar^(2))^(2)+(ar^(2)-a)^(2)+(ar^(3)-ar)^(2)`
`=a^(2)r^(2)(1-r)^(2)+a^(2)(r^(2)-1)^(2)+a^(2)r^(2)(r^(2)-1)^(2)`
`=a^(2)(1-r)^(2){r^(2)+(r+1)^(2)+r^(2)(r+1)^(2)}`
`=a^(2)(1-r)^(2)(r^(4)+2r^(3)+3r^(2)+2r+1)`
`=a^(2)(1-r)^(2)(1+r+r^(2))^(2)=a^(2)(1-r^(3))^(2)`
`=(a-ar^(3))^(2)=(a-d)^(2)`